Priyanka Rao
email: priyankaprakash.rao@mail.utoronto.ca

Student Member, IEEE Chloe Pogue
email: chloe.pogue@mail.utoronto.ca

Student Member, IEEE Quentin Peyron

Member, IEEE Eric Diller

Senior Member, IEEE Jessica Burgner-Kahrs

Modeling and Analysis of Tendon-Driven Continuum Robots for Rod-Based Locking

Keywords: Modeling, Control, and Learning for Soft Robots, Flexible Robotics, Kinematics

come

I. INTRODUCTION

T YPICAL tendon-driven continuum robots (TDCRs) are faced with two major issues due to being compliant and under-actuated: (1) they may lack the stiffness required to perform tasks requiring large force transmissions at their tip and (2) they cannot easily achieve multiple curvatures as they are underactuated, and have limited controllable degrees of freedom.

There have been various approaches to improve the robot stiffness, like granular [START_REF] Cianchetti | Soft robotics technologies to address shortcomings in today's minimally invasive surgery: The stiff-flop approach[END_REF], scale [START_REF] Sadati | Stiffness control of soft robotic manipulator for minimally invasive surgery (mis) using scale jamming[END_REF], and layer jamming [START_REF] Kim | Design of a tubular snake-like manipulator with stiffening capability by layer jamming[END_REF], [START_REF] Langer | Stiffening sheaths for continuum robots[END_REF]. Stiffening by controlling tendon [START_REF] Kim | A stiffness-adjustable hyperredundant manipulator using a variable neutral-line mechanism for minimally invasive surgery[END_REF] and pneumatic actuators [START_REF] Stilli | Shrinkable, stiffnesscontrollable soft manipulator based on a bio-inspired antagonistic actuation principle[END_REF] is another example, where higher tension or pressure is applied on all actuators to increase stiffness. A curvature restraining rod has also been proposed [START_REF] Zhao | A continuum manipulator for continuously variable stiffness and its stiffness control formulation[END_REF] inserted and retracted to vary the robot stiffness. Increasing the controllable degrees of freedom is generally achieved either by adding additional segments or by introducing length extension and retraction in a segment. Varying the tendon routing along the backbone [START_REF] Starke | On the merits of helical tendon routing in continuum robots[END_REF]- [START_REF] Oliver-Butler | Continuum Robot Stiffness under External Loads and Prescribed Tendon Displacements[END_REF] has been shown to change continuum robots' curvature and stiffness properties. Another approach to enable multiple curvatures is by changing the robot stiffness by introducing high friction material [START_REF] Wang | A Dexterous Tip-extending Robot with Variable-length Shape-locking[END_REF] to limit the motion between two sliding backbones in a tip-growing robot.

A promising approach to improving the stiffness while adding additional degrees of freedom in the literature is to use the principle of locking. Locking limits backbone motion as it prevents relative motion between some of the robot components using friction or mechanical principles. The working principle involves having these mechanisms along the robot, clutching or latching on to passive backbone elements, constraining portions of their length. These length constraints result in increased stiffness, and enable multiple curvatures along the backbone and corresponding increase in its degrees of freedom. Compared to the previous methods, locking mechanisms have the advantage of being modular and compact, as they are relatively smaller compared to the robot itself. They can be designed at the millimeter scale, with central channels to pass cables and tools.

A general TDCR design with locking mechanisms that clutch on to additional passive backbones is shown in Fig. 1. These passive backbones run parallel to the primary backbone, with tendons used to actuate the backbone itself. When two locking mechanisms are actuated, the length of the portion

D 1 = {i} τ 1 applied D 2 = {i, j} τ 2 applied D 3 = {i, j, k} τ 3 applied τ 1 τ 2 τ 3 i i j i j k z 1 = (D 1 , τ 1 ) z 2 = (D 2 , τ 2 ) z 3
Base disk Robot disk Locked Unlocked Fig. 2. Sequence of locking motion shown, for m = 3 for a planar case. The notation De denotes the indices of disks whose locking mechanisms are engaged and in locked state while ze denotes the state after a tension τ e is applied. Starting straight configuration with disk i locked, the robot bent on application of τ 1 resulting in state z 1 = (D 1 , τ 1 ). Disk j is then locked as well, constraining the backbones. Tension τ 2 results in z 2 and the state z 2 = (D 2 , τ 2 ). Next, disk k is locked as well and τ 3 applied, resulting in z 3 = (D 3 , τ 3 ).

of the passive backbones between the two mechanisms is constrained. This length-constraint increases the stiffness of the locked portion while allowing the rest of the robot to bend according to tendon actuation.

The locking mechanisms themselves can use a variety of actuation methods. For example, shape-memory alloys (SMAs) have been used to actuate these locking mechanisms to clutch on to rods [START_REF] Yang | Geometric constraint-based modeling and analysis of a novel continuum robot with Shape Memory Alloy initiated variable stiffness[END_REF] and cables [START_REF] Bishop | A Novel Underactuated Continuum Robot With Shape Memory Alloy Clutches[END_REF]. Similarly, in our prior work [START_REF] Pogue | Multiple curvatures in a tendon-driven continuum robot using a novel magnetic locking mechanism[END_REF], a magnetically-actuated mechanical lock was proposed to clutch on to passive backbones along the robot. In an alternative application, these locking mechanisms have been used to create interlaced robots that can achieve followthe-leader deployment. Piezoelectric [START_REF] Kang | The First Interlaced Continuum Robot, Devised to Intrinsically Follow the Leader[END_REF] and parallel-link mechanisms [START_REF] Wang | Follow-the-leader deployment of the interlaced continuum robot based on the unpowered lock mechanism[END_REF] in a robot have been used to clutch on to rods and retain the robot's shape, while another interlaced concentric robot follows the shape.

Modeling the impact of length-based constraints on the robot behaviour is an essential step towards formulating TDCR design, control, and motion planning paradigms. However, existing models typically assume that the segment bends as a constant curvature (CC) arc, which entails certain limiting assumptions. For example, the model proposed by Bishop et al. [START_REF] Bishop | A Novel Underactuated Continuum Robot With Shape Memory Alloy Clutches[END_REF] assumes that the backbone curvature does not vary once locked. Wang et al. [START_REF] Wang | Simplified Kinematics of Continuum Robot Equilibrium Modulation via Moment Coupling Effects and Model Calibration[END_REF] account for this variation by considering moment coupling between the locked and unlocked portion. Additionally, both these approaches assume that the effect of frictional and gravitational forces is negligible to keep the CC assumption valid. A static model is proposed by Yang et al. [START_REF] Yang | Geometric constraint-based modeling and analysis of a novel continuum robot with Shape Memory Alloy initiated variable stiffness[END_REF] that accounts for these forces by assuming a piecewise-constant curvature (PCC) model, where the backbone is represented as a series of constant curvature arcs. However, all the above models assume that the twist in the backbone is negligible and do not account for 3D deformations due to external forces.

In addition to a general static model that can model 3D deformations, there is a need to study the variation in robot properties due to the introduction of locking mechanisms. In particular, there has been no study evaluating the stiffness variation of TDCRs based on the position of these locking mechanisms. Furthermore, most works restrict their study to enabling two curvatures in a single segment.

In this letter, we propose a length constraint approach to model the locked behaviour of these robots. The locked robot portion is modeled by assuming that the length constraints result in an additional internal moment that varies based on the forces acting on the robot. We implement this constraint by using the PCC assumption to model the backbone. The contributions of our letter are as follows. First, we propose a general 3D static model for TDCRs that takes into account the length constraints of locked portions, along with backbone twist, frictional, and gravitational forces. Second, we present a compliance analysis for these locking mechanisms to study the influence of locking mechanism placement on the robot's stiffness. Finally, we propose an actuation strategy using three or more locking mechanisms that can be used to achieve the same shape, but with different stiffness properties.

II. WORKING PRINCIPLE

To model the behaviour of a TDCR experiencing locking, we consider a single segment TDCR actuated by three tendons, routed through n disks with a radius of r d placed equidistantly along the central backbone. Three passive backbones are assumed to be at a distance of r bb to the central backbone in all configurations, also arranged equidistant w.r.t each other. The tendons can have different tensions applied to them which result in different shapes. We illustrate an example TDCR in Fig. 2 as we explain the working principle in the following.

In order to lock any portion of the robot, there needs to be a minimum of two locking mechanisms on the robot. If m is the number of locking mechanisms on the robot, then m ∈ [2, n + 1] ∈ W, since in addition to being placed on the n disks another could also be placed on the base disk. When two locking mechanisms at disks p and q are locked, the length of the secondary backbones between disks p and q is constrained. This locked portion has higher stiffness and can therefore, be used to obtain multiple curvatures when different tendon tensions are applied as shown in Fig. 2.

The shape of the robot depends on the sequence in which the following actions are performed: a) engaging the locking mechanisms and b) changing the tensions applied on the tendons. Let the sequence in which the above steps are performed be denoted by Z = {z 1 , z 2 ...z T }. For any element e, z e = (D e , τ e ), where D e denotes the set of indices of mechanisms which are locked and the vector τ e denotes the vector of tendon tensions applied post locking. If the notation card(D e ) denotes the cardinality or size of D e , the number of locked portions can be calculated by d = card(D e ) -1. If the size of D e is less than 2, no portion is locked as you need at least two locking mechanisms to lock the backbones. From a design perspective, we assume that one locking mechanism is always actuated to prevent the secondary backbones from sliding off.

If mechanisms on disks i, j, and k are locked, D e is denoted as {i, j, k}, where i < j < k, and the portions between i and j, and between j and k is locked. For example, consider the robot with m = 3 locking mechanisms placed at disks {i, j, k} depicted in Fig. 2. First, mechanism at i is locked and a tension τ 1 is applied. Next, the mechanism at j is locked and a tension of τ 2 is applied. Finally, the mechanism at k is also locked and the tension τ 3 is applied.

III. ROBOT REPRESENTATION

In this section we detail the backbone parameterization using the PCC assumption. We follow the nomenclature and methods defined in [START_REF] Rao | How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance[END_REF]. We assume that the backbones do not experience elongation or shear as it is negligible compared to their bending. We consider the model of a single segment robot i.e. there is only one set of tendons actuating the robot, terminating on the last disk. The entire segment is divided further into subsegments, each subsegment j consisting of the portion of the central backbone between disks j -1 and j, including the disk j.

A. Backbone kinematic representation

The point O j-1 describes the center of disk j -1 and represents the base of subsegment j. There is a local frame of reference attached to each disk, with the x-axis of this local frame pointed towards the location of the tendon marked 1, and the local z-axis tangent to the backbone. The tendons and passive backbones are numbered in an anti-clockwise manner as shown in Fig. 3. The segment of length l is divided equally between the n subsegments, each of length l j = l/n.

We assume that the portion of the backbone in each subsegment bends as a circular arc, following the piece-wise constant-curvature assumption [START_REF] Rone | Mechanics modeling of multisegment roddriven continuum robots[END_REF], [START_REF] Yuan | A comprehensive static model of cable-driven multi-section continuum robots considering friction effect[END_REF]. The configuration parameters of the entire segment can then be represented by the parameters, X µ,j = [κ x,j , κ y,j , j ] T for j = 1, 2..n, where κ x,j and κ y,j represent the components of bending curvature along the x and y axis, and j represents the torsional component of curvature. Therefore, 3n parameters (a) (b) are required to represent the entire backbone. The resulting bending curvature, κ j and plane of bending, φ j is given by

O j-1 O j O j 1/ κ j j -φ j x j-1 z j-1 z j x j x j F g + F j-1 j+1 T j,k tj-1 (j-1,j),k T j+1,k tj+1 (j-1,j),k ψ b,1 2 1 3 r bb ψ t,2 3
κ j = κ 2 x,j + κ 2 y,j (1)
φ j = atan2 (κ y,j , κ x,j ) (2)
In a subsegment j with a base at O j-1 and associated reference frame, the coordinates of the end-point of the backbone coincide with the center of the next disk at O j . A point on the central backbone at an arc length of s, defined w.r.t. the frame of reference of disk j -1 is given by

p j-1 j (s) =    cosφj κj (1 -cos(κ j s)) sinφj κj (1 -cos(κ j s)) 1 κj sin(κ j s)    . (3)
The transformation between reference frames j -1 and j is given by

T j-1 j = R z (φ j )R y (κ j l j )R z ( j l j -φ j ) p j-1 j (l j ) 0 1 (4)
where R z and R y represent rotations about the local z and y axis respectively. The transformation matrix between the reference frames of two subsegments is given by sequentially multiplying the corresponding transformation matrices of all disks between the frames of interest.

B. Disk constraints on tendons and passive backbones

The tendons are arranged at angles [ψ t,1 , ψ t,2 , ψ t,3 ] and the passive backbones at angles [ψ b,1 , ψ b,2 , ψ b,3 ] as shown in Fig. 3. The tendons are assumed to be partially constrained and the coordinates of tendon k intersecting at disk j, w.r.t. frame j is calculated by

p j t,(j,k) = r d cos (ψ t,k ) sin (ψ t,k ) 0 (5)
The coordinates of the passive backbone k can be obtained from ( 5) by substituting ψ t,k with ψ b,k , and r d with r bb .

C. Coordinates of points along passive backbones

Since we consider that the length of the locked portion of the passive backbones, consisting of multiple subsegments, remains constant, we need to calculate their length within a subsegment. To do so while considering their experienced twist, we approximate a continuous secondary backbone by a series of w intermediate points. Since we assume that a subsegment has constant torsional curvature, the twist angle at an arc length of s ∈ [0, l j ] in the subsegment is given by j s. Using equations ( 3), ( 4) and ( 5), the corresponding point lying on backbone k w.r.t. the frame j -1 is given by

p j-1 b,(j,k) (s) = R j-1 (s) p j-1 j (s) 0 1 p j b,(j,k) 1 (6) R j-1 (s) = R z (φ j )R y (κ j s)R z ( j s -φ j ) (7)
The coordinates of the w points can be obtained by equating s = (m/w)l j , where m = 1, 2...w.

IV. STATIC MODEL

We consider a tension τ k applied at the tendon k to actuate the robot. The static model involves finding the mapping from these applied tensions to the resulting position and orientation of the entire backbone. We assume that the tendons follow a partially constrained path, defined in [START_REF] Rao | How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance[END_REF], where the portion of the tendon between consecutive disks is considered to be a straight line. The force equilibrium equations have been adapted from the model written for a standard TDCR design, proposed in [START_REF] Yuan | A comprehensive static model of cable-driven multi-section continuum robots considering friction effect[END_REF]. While the above methods have been used to model the unlocked robot, we introduce additional length constraints on the portion of the locked passive backbones to model the robot in the locked state. Once the force and moment equilibrium equations are written for each subsegment, the Hooke's law is used to find the resulting constitutive equations. The backbone is assumed to be inextensible, and its weight negligible compared to the weight of the disks.

A. Forces and moments

The tendon interaction forces, described in [START_REF] Rao | How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance[END_REF] in each subsegment j due to the kth tendon is denoted by F j-1 j,k , and the resulting moment by M j-1 j,k . In addition, there are external and gravitational forces acting on the subsegment with resultant moments. Both these forces must be defined w.r.t. the local frame of reference as

F j-1 g = T 0 j-1 -1 0 0 -m disk,j g 0 (8) F j-1 ext = T 0 j-1 -1 F ext,j , (9)
where g is the acceleration due to gravity, F ext,j is the external force applied to disk j, if any, expressed w.r.t. the global frame of reference. The mass of disk j is denoted by m disk,j and includes the additional mass of the locking mechanism, if any. If there is no external force applied to that subsegment, the magnitude of F ext is set to zero. Using the equilibrium equations described above and equating them as done in [START_REF] Rao | How to Model Tendon-Driven Continuum Robots and Benchmark Modelling Performance[END_REF], the net force and moment in a subsegment, F j-1 j and M j-1 j can be calculated.

Using the constant curvature assumption for each subsegment and Hooke's law, the resulting moment due to bending of the central and passive backbones is given by

M j elastic = R(φ, θ j )     0 E(I c κ j + 3 k=1 I b κ j,k ) G(J c + 3 k=1 J b ) j 0     (10) R(φ, θ j ) = R z (φ)R y (θ j ) (11)
κ j,k = κ j 1 -κ j r d cos(ψ b,k -φ j ) (12)
where I c , I b , J c , and J b are the second moment of area, and polar second moment of area of the central and passive backbones respectively. Assuming that the central and passive backbones are made of the same material, E and G are the Young's and Shear modulus respectively.

B. Constitutive equations

In addition to the internal moments generated, we assume that each locked portion of the backbones experiences a corresponding internal moment due to the locked state. For a configuration represented by z e = (D e , τ e ), if the set D e consists of d elements, then d unknown internal moments, M int,i , ∀i = 0, 1, ..d are considered to be acting on the subsegments in the locked region.

The function g(j) indicates whether the subsegment j is locked or not.

g(j) = 1 if locked 0 else. ( 13
)
The net internal moment on subsegment j can then be calculated by summing the moments given in [START_REF] Oliver-Butler | Continuum Robot Stiffness under External Loads and Prescribed Tendon Displacements[END_REF] and the sum of internal moments resulting from being a part of any of the d locked portions. Transforming the latter to the local frame, the net moment is given by

M j-1 j = M j elastic + g(j)T j-1 j M j int,i (14)
At equilibrium, the internal moment must be equal to the net moments resulting from force interactions on the robot. Therefore, there are 3n equilibrium equations due to moment balance at all n subsegments.

C. Passive backbone length constraints

We can obtain 3d additional constraint equations by considering the fixed lengths of the three passive backbones. For any locked portion i between disks p and q, the length of the secondary backbones between them, where p < q can then be calculated using (6) as

p,q k = q j=p+1 w m=1 p j-1 b,(j) ml j w -p j-1 b,(j) m + 1 w l j (15)
where ||.|| represents the Euclidean norm. For ease of notation, the subscript k is removed from the above equation. If the locked portion between p and q represents the ith locked portion out of d locked portions, i is the vector representing the lengths of the passive backbones in it. Once locked, these lengths must remain the same when the robot is bent to another shape by applying a different set of tensions. This constraint can be expressed as

i - * i = 0, ∀i = 0, 1, 2, ..d (16)
where * i denotes the backbone lengths of the locked portion predicted by the model after the tendon tensions are changed.

D. Discussion

In the locked state, the system now has 3n + 3d equilibrium equations from the moment balance of internal and applied moments. There are also 3n + 3d unknown parameters, consisting of the 3n curvature parameters as well as the three components of d internal moments generated. When there are no locked portions, the model reduces to that of a general TDCR with 3n unknowns and 3n constraint equations.

To calculate the final shape at the end of a sequence Z = {z 1 , z 2 , ...z T }, the model needs to be run successively for each preceeding step. Doing so allows us to 1) calculate the lengths of the locked portions of the passive backbones, and 2) use the curvature values as the model's initial guess to aid convergence when evaluating the next step. The models are computed in MATLAB, using the Trust-region-dogleg algorithm implemented in the fsolve function.

V. EXPERIMENTAL VALIDATION

We investigate the performance of our proposed model for different locked configurations of a single segment TDCR. We specifically look at the evaluation of m = 2 and m = 3 locking mechanisms. The three different case studies considered to evaluate the model and its results are described first. The experimental setup and calibration process are then described, followed by the evaluation.

A. Case studies

Typically, TDCRs with on-board locking mechanisms only consider two mechanisms placed at the base (disk 0) and an intermediate disk. We extend the study to two additional cases as well, considering external tip forces. The robots' final shapes in all three cases are used to evaluate the model.

1) Case 1, (m=2):

The locking mechanisms are placed at {0, 8}. First, disk 0 is locked and the robot is bent using a tension τ 1 . Next, the locking mechanism at 8 is locked and another tension τ 2 is applied. The sequence of locking is given by Z 1 = {({0}, τ 1 ), ({0, 8}, τ 2 )}.

2) Case 2, (m=2): Here, locking mechanisms at {8, 16} are considered such that the distal portion of the robot is locked. The sequence of locking is given by Z 2 = {({16}, τ 1 ), ({8, 16}, τ 2 )}.

3) Case 3, (m=3): Three locking mechanisms placed at {0, 4, 12} are considered. The sequence of locking is given by Z 3 = {({0}, τ 1 ), ({0, 4}, τ 2 ) ({0, 4, 12}, τ 3 )}.

B. Experimental setup and calibration

To evaluate the three cases in section V-A, two robot prototypes were developed. Both robots are 240 mm long, are actuated by three tendons arranged around the robots' central backbone at angles of ψ t = [0, 2π/3, 4π/3], and have three passive backbones placed at angles of ψ b = [π/3, π, 5π/3]. Both the passive backbones and tendons are equidistant from the central backbone, with r d = r bb = 4.25 mm. The first robot had locking mechanisms at disks 0, 8, and 16, which could be selectively locked to achieve cases 1 and 2. The second robot had locking mechanisms at disks 0, 4, and 12, and was used for case 3. The locking mechanisms used in these prototypes used set screws for locking. These set screws were manually tightened, engaging them with the robots' passive backbones. While set screw-based locking mechanisms were used in this experiment for simplicity, they can be replaced with any type of locking mechanism, and the model will still be valid. The two robots were actuated by hanging calibrated weights on the tendons.

The desired 2-or 3-curvature robot configurations were achieve by sequentially engaging the robots' locking mechanisms and applying tension to their tendons as discussed in section V-A. The locking sequence for case 3 is shown in Fig. 4. The values of tendon tensions were chosen randomly between 50-450g calibrated weights, with an additional weight of 50 g used to pretension all the three tendons.

Once the final shape of the robot was achieved (two curvatures in cases 1 and 2, and three curvatures in case 3), a coordinate measurement machine with a point probe (Microscribe G2X, Revware Inc., USA, accuracy of 0.002 ") was used to measure the position of every fourth disk on the robot (Fig. 5). Six points were measured around the edge of each fourth disk which were used to find the position of the robot's backbone at that disk. In addition to evaluating the robot in free space, we evaluated the performance of the model for a force acting on the tip at the end of the sequence for cases 1 and 2. Calibrated weights were hung from the end disk using a cable, using a random calibrated weight ranging from 2 g -5 g. Since the configurations are not restricted to the planar case, the applied weight results in nonplanar deformations with twists.

For the two prototypes, we use a sample of 25 of the 60 readings to optimise for the Young's modulus of the Nitinol rods as well as the orientation of the base. The latter is expressed as rotation of the experimental data about the global x, y, and z axis. The calibration is posed as a least-squares optimization, minimizing the average Euclidean distance between observed and predicted tip positions. The optimisation was solved using the Nelder-Mead Simplex method, implemented by Matlab's fminsearch function.

C. Evaluation of proposed models performance

The model is first evaluated for a TDCR operating in free space. For all three cases, the values of tensions are varied to result in different 3D configurations. A total of 40 configurations are measured and the performance of the model on this dataset is tabulated in Table I. The average end-effector (disk 16) position error is 3.67 % (8.8 mm) w.r.t. total length.

The model was evaluated with an applied tip force over 20 observations presented in Table II. The average position error for the end disk is 3.61 % w.r.t. length for the 20 observations (8.7 mm). The observed errors in the experiments is consistent with the average error of 2.67% observed by Gao et al. [START_REF] Gao | A general friction model of discrete interactions for tendon actuated dexterous manipulators[END_REF] for a robot of length 79.2 mm when modeled with a general friction model for TDCRs. The reported error could be further reduced with a more accurate model of the observed friction, such as that between the passive backbones and disks, but that is currently beyond the scope of this letter. We observe that the errors are higher for case 3 as it has a larger number of actuation steps, with accumulated manual error at each step.

VI. COMPLIANCE ANALYSIS

The effect of the position and number of locking mechanisms on the stiffness of a single segment TDCR is analysed in this section. We assume the effect of gravitational and frictional forces to be negligible in the studies below to study the behaviour of the locking independent of these factors. We consider a robot with 16 disks, with three locking mechanisms. Two of the locking mechanisms are fixed at disks 0 and 16, whereas the third one is placed on disk d, where the value of d is varied such that d ∈ [1, 2, ...15].

A. Case Studies

We analyse two configurations achieved through two different sequences of locking :

• Configuration 1 (proximal portion locked): Locking mechanism at disks {0, d} locked with Z 1 = {({0}, τ 1 ), ({0, d}, τ 2 )}. • Configuration 2 (distal portion locked): Locking mechanism at disks {d, 16} locked with Z 2 = {({16}, τ 1 ), ({d, 16}, τ 2 )} When the effects of gravity and friction are considered negligible, the shape predicted by the model of the two portions is decoupled. In such a case, a TDCR with n locking mechanisms behaves as an n-segment TDCR. The workspace, forward, and inverse kinematics can be obtained from results already derived for these robots. The more significant difference likely arises from stiffness properties as length constraints imposed by the locking change the inernal stiffness. Therefore, in this section, we study the compliance of the robot using the definition of compliance manipulability index (CMI) in [START_REF] Khadem | Force/Velocity Manipulability Analysis for 3D Continuum Robots[END_REF].

The CMI is the volume of the compliance manipulability ellipsoid, and is the product of the singular values of the compliance matrix evaluated at the tip of the robot. The compliance matrix itself is evaluated by measuring the change in tip pose per unit wrench applied at the tip. The CMI is always greater or equal to zero. Higher value of CMI indicates a lower stiffness, and a value 0 denotes infinite stiffness.

If we disregard shapes where the entire backbone is locked, there are two major design choices that can be made for the above. One, we can design the system to only be proximally or distally locked (can only achieve configuration 1 or 2). In such a case, only one locking mechanism is required and the backbones can be permanently fixed at either the base or the end disk (as done in multibackbone robot designs [START_REF] Simaan | Design and integration of a telerobotic system for minimally invasive surgery of the throat[END_REF]). Two, the three locking mechanisms can be used as described above, alternating between proximal and distal section being locked (can achieve both configuration 1 and 2). Since the first design choice is a subset of having three locking mechanisms, we study the second design choice in following subsections. First, we propose an actuation strategy in subsection VI-B that can help us leverage the use of three locking mechanisms.

Next, we study the variation in robot stiffness or CMI for both configurations in subsection VI-C.

B. Actuating all three locking mechanisms: Strategy to obtain the same robot shape with different compliance

If the secondary backbones are permanently fixed at either the base or the end disk (as done in multibackbone robot designs [START_REF] Simaan | Design and integration of a telerobotic system for minimally invasive surgery of the throat[END_REF]) but with an additional locking mechanism on an intermediate disk, the robot behaviour can be modeled as that of either Configuration 1 or 2 respectively. Based on the Configuration chosen, the robot will have different stiffness. However, if three locking mechanisms are used (one at the base, one at the end disk, and one along the robot body) instead of permanently fixing the backbones, we can leverage their locking ability to obtain the similar backbone curvatures but with different stiffness as shown in the following subsection.

The actuation strategy is as follows. We observe that when the magnitudes of tendon tension are interchanged in Configuration 2, i.e. τ 2 is first applied, and then τ 1 , we obtain the same backbone curvatures. First, the distal section gets locked in a curvature resulting from τ 2 , and then the proximal section is bent due to τ 1 , resulting in the same shape as configuration 1 as shown in Fig. 6. The two different sequences would be Z 1 = {({0}, τ 1 ), ({0, d}, τ 2 )} resulting in configuration 1, and Z 2 = {({16}, τ 2 ), ({d, 16}, τ 1 )}, resulting in configuration 2.

This property can be exploited to achieve the same backbone curvatures but with different stiffness properties. The strategy can be extended to more than three locking mechanisms by following the same principle. If the sequence in which tension is applied is reversed, and the disks are locked in decreasing magnitude of their index, starting with the end disk, the same robot shape can be obtained. Theoretically, we could switch between the two configurations with instantaneous changes in tendon tensions and locked states of the disks. However, as mechanical components cannot be instantaneously actuated, vibration and robot motion are to be expected.

C. Actuating two locking mechanisms: Variation in robot compliance

This subsection studies the variation in robot compliance when proximally or distally locked with varying values of the location d of the third locking mechanism. First, the robot is bent to a C-shape and then the locking mechanism at disk d is locked resulting in either the proximal or the distal portion being locked. Without changing the tendon tension, the robot compliance is calculated. The value of tendon tension is approximated as τ 1 = τ 2 = [τ, 0, 0] where τ = π 2lr d E(I c + 3I b ) to estimate the tension required to achieve 90 degree bending of the robot. This value is calculated using the Euler-Bernoulli beam theory by assuming that the effective flexural rigidity of the primary and secondary backbones is given by E(I c +3I b ). The variation of CMI with the index of the locked disk, d is plotted in Fig. 7 (a). The overall compliance in both cases reduces compared to a single segment TDCR with no locked portion, which has a CMI of 1.36 when bent to a C-shape. Therefore, we see that the robot is more compliant in the absence of locking mechanisms.

(Proximal locked) Configuration 1 (Distal locked) Configuration 2 τ 1 τ 2 τ 1 τ 2 τ 2 τ 1 (a) (a) (b) (b) (c) (c) (d) (d)
Next, we study the variation of stiffness when the robot is bent to an S-shape. We apply the strategy described in the subsection VI-C, where the values of τ 1 and τ 2 are interchanged to achieve the same configuration. For configuration 1 (proximally locked), the value of τ 1 = [τ, 0, 0] is retained, and τ 2 = 2[0, τ, τ ] to ensure that the S-shaped curve is formed. The values are interchanged for configuration 2. The observed CMI are plotted in Fig. 7(b).

We observe that the difference in compliance between proximal and distal portions being locked is not significant when the length of the locked portion lies approximately between 30 -70%. As the length of the locked portion decreases, the length of the deforming unlocked portion increases, resulting in a more compliant robot. The compliance reduces sharply when the length of the locked portion greater than approximately 20 -25% of the entire length. Comparing the robot compliance between S-shape and C-shape configurations for given locking disk index, the largest differences are observed for Case 1 when the index is below 5. This difference is because the length of the locked portion is smaller than the free moving portion, which then bends to form an S-shape that differs widely from the corresponding C-shape. The compliance otherwise doesn't vary significantly because the resulting shape is similar in both cases to a C-shape. All the above could inform the placement of the locking mechanism based on the stiffness requirements as well as advise design choices when implementing the strategy proposed in subsection VI-B.

VII. CONCLUSION AND FUTURE WORK

In this work we proposed a static model for a TDCR with locking mechanisms that can account for variation in curvature along the backbone while considering frictional and gravitational forces. The backbone curvature representation includes twist, which allows for the modeling of 3D deformations. In addition, we demonstrate that locking can be used to modulate the robot stiffness without changing its shape using an appropriate actuation strategy. The design analysis highlights the variation of robot compliance with placement of locking mechanisms. Based on the task's compliance requirements, the location and number of locking mechanisms can be determined. While we assume that the passive backbones run parallel to the central backbone and do not experience friction at disks, the modeling can be further improved by accounting for their buckling and frictional forces. In the current work, we analyse the performance of the robot for up to three locking mechanisms. Future work will benefit from analysing the design choices and consequences of adding additional mechanisms on the robot properties. The proposed strategy to achieve different stiffness can inform their motion planning and will be investigated in future work.

Fig. 1 .

 1 Fig. 1. Sequence of locking motion shown, with the multiple curvatures

Fig. 3 .

 3 Fig. 3. (a) Diagrammatic representation of the forces acting on the robot marked in red and coordinate axes in orange. (b) Cross section of the disk showing the anti-clockwise numbering of tendons and backbones. The passive backbones are placed at a radius of r bb , marked in green, while the tendons are placed at a radius of r d , marked in black.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Locking sequence for a continuum robot with three locking mechanisms placed at 0, 4, 12 for Case 3. (a) The locking mechanism at disk 0 is locked and tension τ 1 is applied. (b) The locking mechanism at disk 4 is locked and tension τ 2 is applied. (c) The locking mechanism at disk 12 is locked and tension τ 3 is applied. The sequence of locking is given by Z 3 = {({0}, τ 1 ), ({0, 4}, τ 2 ) ({0, 4, 12}, τ 3 )}.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Strategy to obtain the same backbone curvature through two different locking sequences, with Configuration 1 (proximal section locked) in green boxes and Configuration 2 (distal section locked) in purple boxes. From the straight configuration in (a), the robot is bent by applying a tendon tension of τ 1 for 1 and τ 2 for 2, shape (b) is obtained. Once the proximal and distal section is locked in (c), by applying antagonistic tendon tension, shape (d) is obtained, with the same backbone curvatures in both sections.

TABLE I ERROR

 I STATISTICS FOR DISKS 8 AND 16, EXPRESSED AS % W.R.T LENGTH FOR THE THREE CASES OPERATING IN FREE SPACE FOR ROBOTS OF LENGTH 240 mm. AND 16, EXPRESSED AS % W.R.T LENGTH FOR CASES 1 AND 2, EXPERIENCING DIFFERENT TIP FORCES OF MAGNITUDE 2 -5 G FOR ROBOTS OF LENGTH 240 mm.

		Case 1 (Z 1 )	Case 2 (Z 2 )	Case 3 (Z 3 )
		(15 readings)	(15 readings)	(10 readings)
	Disk	8	16	8	16	8	16
	Average	1.75	3.18	2.45	3.01	2.70	5.38
	Median	1.83	2.70	2.32	3.23	2.50	3.63
	Minimum	0.45	1.19	0.55	0.98	0.76	1.42
	Maximum	2.97	5.96	4.05	6.77	4.26	11.08
			TABLE II
	ERROR STATISTICS FOR DISKS 8 Case 1 (Z 1 )	Case 2 (Z 2 )
			(10 readings)	(10 readings)
		Disk	8	16	8	16
	Average	3.00	3.22	1.83	3.99
	Median	2.87	2.69	1.79	3.76
	Minimum	1.41	1.18	0.54	1.92
	Maximum	4.58	6.38	3.60	6.67

This paper was recommended for publication by Editor-in-Chief T. Asfour and This work was supported by the XSeed funding program of the Faculty of Applied


Mathieu Schumann

Quentin Reynaud

François Sempé

Julien Guibourdenche

Jean-Baptiste Ly

Nicolas Sabouret

A multi-sourced data and agent-based approach for complementing Time Use Surveys in the context of residential human activity and load curve simulation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The world of energy has been undergoing structural changes notably associated with the electrification of energy uses and the insertion of local renewable energies into smart grids. The increased understanding of the challenges of climate change as well as the ongoing energy crises are also driving the need to accelerate the energy transition in the building sector. Indeed, buildings still account for 30% of the global final energy consumption in 2021, according to the IEA. However, this can only be achieved with the involvement of citizens at both the individual and collective scales. Performing human-centric assessments of energy consumption at the household and urban levels (Dabirian et al., 2022) is critical to make these changes operational. The now-documented impact of human activity on energy consumption in buildings brought the research community together to tackle the various scientific issues arising from this topic, particularly within IEA EBC Annex 79 [START_REF] Dong | Occupant behavior modeling methods for resilient building design, operation and policy at urban scale: A review[END_REF]. An increasing number of approaches use Time Use Surveys (TUS) and stochastic individual-based approaches to generate occupancy profiles, model interactions with building systems, or evaluate energy consumption related to human activity. However, the state-of-the-art analyses [START_REF] Osman | A comprehensive review of time use surveys in modelling occupant presence and behavior: Data, methods, and applications[END_REF] identify limitations to using TUS to simulate coherent human activity and the associated energy consumption [START_REF] Yamaguchi | A cross analysis of existing methods for modelling household appliance use[END_REF]). In the first section, we review and discuss these limitations. The second section presents our proposition to complement TUS data to effectively address these limitations. We show how qualitative studies on human activity in social sciences and cognitive ergonomics are essential to complement TUS data that are purely quantitative. Then we describe our human activity model based on Agent-Based Modelling (ABM) and how this paradigm allows us to tackle the discussed TUS limitations. We describe how this led to the improvement of the appliances use and energy consumption model and illustrate its validation process, with a focus on specific electricity. Finally, we demonstrate the practical applications of our model through two Use Cases. [START_REF] Mckenna | Simulating residential demand response: Improving socio-technical assumptions in activitybased models of energy demand[END_REF] as well as [START_REF] Osman | A comprehensive review of time use surveys in modelling occupant presence and behavior: Data, methods, and applications[END_REF] list major issues regarding TUS data and their use. We summarize and discuss them in the following list: Limitation 1 (L1): no link between activities and energy use Classical state-of-the-art approaches tend to oversimplify the relation between appliances and activities (e.g., by directly linking appliances to activities), leading to inaccurate and too static temporal appliance use profiles. This relation is highly individual-dependent and totally absent in TUS data. For example, the "laundry" activity in TUS data could be done with or without a washing machine; and the relation between the "cooking" activity and the concrete use of an oven is complex and impossible to model solely based on TUS data. TUS also fail to capture any potential energy consuming situations that are not directly linked to the current activity ("watching TV while eating" or "having a radio on in another room"). L2: lack of data about variability and interactions between activities TUS data lack explanation about sequences of activities and days because it tends to be collected for single days only. More generally, TUS data fail to capture the temporal activities variability: routine activities, temporal variations around routine activities, and exceptional activities. Classical state-of-the-art approaches tend to use these data to represent average days, neglecting the behavioral change due to external factors (weather, communal events, national holidays, etc.) as well as internal factors (mental or physical states, emotions, etc.). They usually fail to produce a consistent series of activities because they assume that activities are random variables that depend on a fixed number of independent variables (e.g., day of the week, total number of residents in a household, etc.) that do not vary in time. [START_REF] Mckenna | Simulating residential demand response: Improving socio-technical assumptions in activitybased models of energy demand[END_REF] point out that activities are interdependent and that time-shifting one can have complex impacts on others. They stress that understanding interdependencies and evaluating the impact of changing activity timings is crucial for demand response studies. The absence of variability in activity patterns is a significant concern for modeling human activity, as it is a crucial feature according to [START_REF] Attaianese | Human factors and ergonomic principles in building design for life and work activities: An applied methodology[END_REF]. L3: lack of data about the impact of individual characteristics of occupants, households, dwellings and appliances TUS-based approaches tend to average individuals and neglect factors that may have strong associations with energy demand (employment status, work patterns or whether children, elderly relatives, guests). The characteristics of an average resident may have little meaning for demand response purposes if few people, in practice, conform to the characteristics of the average. Similarly, appliances are often assumed to be average both in terms of technical characteristics and use which decreases the diversity of energy consumptions. This results in the simulation of households that are overly representative of the average, whether in terms of household members or their dwellings. L4: absence of collective activities or household-level organisation TUS data are focused on the individual level and makes it hard to detect the overlapping between household members while performing shared activities. Moreover, TUS data assume that residents perform one activity at a time and that activities are not combined (e.g., a continuous combination of childcare and watching TV is not a recognized state). However, activities and consumptions are the product of individual, collective and interdependent practices [START_REF] Shove | What Is Energy For? Social Practice and Energy Demand[END_REF] resulting from complex household dynamics, but this higher level of organization is often overlooked in stateof-the art approaches [START_REF] Yamaguchi | A stochastic model to predict occupants' activities at home for community-/urban-scale energy demand modelling[END_REF]. L5: biases in TUS Although TUS are using a proven methodology, they exhibit some major biases. Firstly, all short activities (<10 minutes) may have trouble of being captured in TUS data since TUS standards rely on 10-min episodes. "Boiling water with a kettle" or "opening a window" may be totally absent from TUS datasets. Moreover, [START_REF] Osman | A comprehensive review of time use surveys in modelling occupant presence and behavior: Data, methods, and applications[END_REF] underline the fact that TUS surveys are affected by the respondents' memory, accuracy, and willingness to round up or down the actual time spent on different activities, as well as by the social desirability and bias that might overestimate the time spent on some activities rather than other. In addition, these purely quantitative data tend to hide the fact that a single activity does not mean the same thing for different people [START_REF] Poizat | Analysis of activity in domestic settings for the design ubiquitous technologies[END_REF]. "Gardening" could be a job or a leisure activity; "cooking" could be a solitary, time-pressed activity, considered a chore, or a collective and relaxing activity. Furthermore, TUS surveys are generally conducted every 10 years (but often less than that), which raises the question of the impact of potentially outdated data.

State of the art: limitations of TUS data and their use

State-of-the-art propositions to tackle these limits

The literature reports numerous works that propose advancements regarding some of these limitations. For example, [START_REF] Foteinaki | Modelling household electricity load profiles based on Danish time-use survey data[END_REF] propose a model to associate activities and appliances based on percentages of appliance ownership at a national level (related to L1). [START_REF] Yilmaz | Occupant Behaviour Modelling in Domestic Buildings: The Case of Household Electrical Appliances[END_REF] introduce an empirical-based stochastic model of appliance use that benefit from monitoring spanning over several days (related to L2). Buttitta et al. (2019) proposed a multi-day occupancy model based on a Markov Chain process to generate weekly heating-load profiles (related to L5). But to our knowledge there is no publication that presents a more consistent approach to all the TUS limitations. In line with [START_REF] Schumann | Multi-Agent based simulation of human activity for building and urban scale assessment of residential load curves and energy use[END_REF], who pointed out some issues related to classical occupant behavior models, Osman et Ouf (2021) highlighted the potential of integrating TUS with different sources of data that cover social, economic, and building aspects. They recommended to integrate TUS data with energy metered or surveyed data to fill any missing information and get a holistic view for the occupants' behavior and their impact on energy use profile. In addition, Berger and Mahdavi (2022) suggest that ABM can effectively capture the behavior of building occupants, both as individuals and groups. However, they note a deficiency in studies that examine the incorporation of ABM in building performance simulation. The authors underline the usual limitations associated with ABM: lack of detailed information about agents' behavior and their interactions, reproducibility issues, and difficulty to evaluate the fidelity of ABM results given their potential emergent complexity. This paper proposes to answer these criticisms through a comprehensive approach linking qualitative and quantitative studies on human activities with ABM to address the simulation of energy consumption in the residential sector.

Our combined approach of qualitative and quantitative studies and ABM

The focus of this paper is our approach to employing and supplementing TUS data (whereas a general presentation of our platform can be found in [START_REF] Schumann | Multi-Agent based simulation of human activity for building and urban scale assessment of residential load curves and energy use[END_REF]. This work is rooted in a technological research program that considers conceptual, methodological, and technical aspects, including the relationship between activity models and multi-agent systems, qualitative and quantitative validation, and the development of a multiagent simulator platform. First, we explain how qualitative empirical studies are essential for understanding human activity and tackling the aforementioned TUS limitations. The need for qualitative and situated approaches to human activity and ABM Qualitative studies of human activity favor conceptual and methodological frameworks that enable a detailed understanding of 'how' individuals and groups act in concrete everyday settings (e.g., [START_REF] Guibourdenche | Concerns and arrangements in contexts of domestic activity. Contribution to the design of energy-efficient and appropriable ubiquitous computing environments[END_REF]. Therefore, the SMACH project initially relied on the structural relationship between the qualitative studies of human activity and ABM to build its approach to activity modeling and its validation [START_REF] Haradji | Human activity and social simulation[END_REF]. This first step notably meant articulating concepts, methods, and results from qualitative studies of activity in natural settings with incremental-situated ABM and social simulation. This work produced a basis for small-scale simulation (a few hours for one single household). This approach helped building coherent Agent Based (AB) sequences of individual actions in interaction with the other members of the family and the environment, at the household scale. It only then became apparent that TUS were necessary when we needed to work at a larger scale (thousands of households over several months of simulated time). Qualitative studies allow domain experts to understand and explain the various forms, structures, dynamics, emergence, construction, or meaning of in situ activity. A wide range of approaches to human activity exists in the social sciences and ergonomics. These works require various qualitative methods, e.g., eliciting the action's meaning, observing practices, and analyzing conversations or verbalizations. The situated ABM approach is partly inspired by these frameworks, while being a reduction of activity analysis for ABM. It consists in placing a human participant in a simulated situation and observing their responses instead of collecting general information through an interview. The ABM is adjusted through a step-by-step approach based on the knowledge of the participant's real activity [START_REF] Sempé | Incremental and Situated Modeling for Multi-Agent Based Simulations[END_REF]. Derived from these studies, three principles formed the qualitative basis for the SMACH project: 1) the individual agent are autonomous in realizing their tasks, 2) the collective activity is built in the dynamic interaction between the individuals (forming small and ephemeral sub-groups in the home), and 3) there is an asymmetric structural coupling between individuals/groups and their environment (devices, building, outdoor conditions). This approach provides solutions to the TUS limitations outlined in the first section. Concerning L1 ("no link between activities and energy use"), the structural relationship between qualitative studies and ABM can help make accurate links between activities, devices, and energy use. [START_REF] Guibourdenche | Concerns and arrangements in contexts of domestic activity. Contribution to the design of energy-efficient and appropriable ubiquitous computing environments[END_REF] empirically described the various contexts of inhabitants' activity (individual, collective, and coupling to the material environment) in which the devices are used or not, forgotten or intentionally left on. He formally described the activity contexts of energy consumption by focusing on the parallel and interdependent concerns of individuals in action at home, e.g., a mother ironing and watching TV while coordinating the activity of the rest of the family in order not to be late for upcoming children's baths and the rest of the evening. This kind of study provides qualitative arguments when simulating the variations of activities, along with situated modeling and social simulation. It aligns with [START_REF] Shove | The dynamics of social practice[END_REF]'s notion of practices as building time interdependently and with the approach considering "what energy is for" [START_REF] Shove | What Is Energy For? Social Practice and Energy Demand[END_REF]. Both standpoints of qualitative studies of action and situated ABM imply relating L2 (lack of variability and interactions) and L5 (TUS's biases): we need to use concepts and methods that enable us to describe the variety (richness) of action meaning and the individualhousehold-environment interactions. Qualitative approaches consider temporal scales of activity ranging from a tenth of a second to several years. They are not limited to TUS's 10 minutes episode. Capturing short or more extended activities depends on the researchers' choice or the modeler's choice in situated modeling. As a result, they help characterize and explain activity's variations in real life. In a study focusing on second-to-day long activities, [START_REF] Guibourdenche | Concerns and arrangements in contexts of domestic activity. Contribution to the design of energy-efficient and appropriable ubiquitous computing environments[END_REF] shows how a mother moved the pile of laundry to be ironed the evening before this ironing (on a Wednesday) from the first floor of the house to the sofa in the living room on the ground floor, thus preparing for the ironing planned for the next day (Thursday). The next day (Thursday), in a period from 4:00 p.m. to 5:57 p.m., the processing of the clean laundry carried out by this same person is suspended 18 times. These suspensions are explained by the other actions and concerns to be realized by the mother, particularly caring for the children. These kinds of results helped frame explanations and produce coherent sequences for ABM, although the AB model is a substantial reduction of real-life activity. L4 ("absence of collective activities or household-level organization") is another area where qualitative studies, situated modeling and social simulation have already described many phenomena concerning household energy consumption and social organization. As an illustration, a model exists for considering the precise degree of collective similarity and convergence between individual concerns in a household [START_REF] Haradji | From modeling human activity to modeling for social simulation: between realism and technological innovation[END_REF]. Situated modeling and social simulation help frame and adjust AB models to household dynamics in real life given technical possibilities and limits for reduction into a model. For example, [START_REF] Sempé | Incremental and Situated Modeling for Multi-Agent Based Simulations[END_REF] demonstrated how to create variations in the individual or collective realization of breakfast. L3 remains a more practical issue, depending on the researcher's choices about the population to be studied. Nevertheless, studying large categories from a situated and qualitative perspective on activity would call for further research. Although further research is still necessary, several of the TUS limitations identified by [START_REF] Mckenna | Explaining daily energy demand in British housing using linked smart meter and sociotechnical data in a bottom-up statistical model[END_REF] have been addressed in the SMACH project with the help of this structural relationship between ABM and the qualitative studies of real human activity: activities in parallel (individual or collective); short activities; consumptions as the indirect effect of collective and interdependent activities resulting from complicated household dynamics; coherent sequences of activities; explanation about sequences of activities and days. If "the concern is now not just who, but what is acting, and how" (Malik et al., 2022), qualitative and situated approaches to human activity and ABM can provide new opportunities for ABM and better comprehension of energy consumption beyond the limits of TUS. The main challenge remains to combine the concepts, methods, and results of qualitative and quantitative approaches to human activity and energy.

A human activity model based on ABM

We now introduce the agent model and how it benefits from multi-source qualitative and quantitative data. The model we present consists of two distinct modules: a population generation module and an activity module. The population generation module, based on a classical model from the synthetic population research field [START_REF] Müller | Population synthesis for microsimulation: State of the art[END_REF], allows the generation of a synthetic population that exhibits the same characteristics as a targeted population. Based on large scale national studies (e.g. the "2014 French population census" and the "2017 housing survey" from the French National Institute of Statistics and Economic Studies), the module generates up to tenths of thousands of individuals (characterized by their gender, age, PCS, income, etc.), gathered in households (characterized by their family type, size, energy tariff, etc.), installed in dwellings (characterized by their surface, type, insulation, localization, weather), and equipped with appliances (heaters, water heater, light bulbs, home appliances, electric vehicle, solar panels, etc.). These features are inter-dependent and will impact the simulated load curves. For instance, the household's size is related to the dwelling's floor area, and both impact the hot water tank volume, whereas the localization affects the presence probability of an air conditioner. The population generation module opposes L3 directly. The activity module notably addresses L2. In the best cases, TUS data contain 2 days for the same individual, but these days are non-consecutive and of a different type: a weekday and a weekend day. This makes it impossible to infer how much an individual acts similarly (or dissimilarly) from day to day. [START_REF] Yamaguchi | A stochastic model to predict occupants' activities at home for community-/urban-scale energy demand modelling[END_REF] use the notion of "routine" and "non-routine" behavior, which offers a promising approach to address L2. However, further data are required to support this model, specifically in terms of establishing criteria for distinguishing between routine and non-routine behaviors. State-of-the-art approaches have two classical options to deal with TUS data: either real timetables are directly copied and simulated (which leads to fixed scenario patterns), either these data are aggregated by "types of individual" (e.g. by sex, age, PCS, income, etc.) to represent their "average days". Both cases logically lead to a severe lack of variability of activity. We propose to overcomes this impasse by adopting a third alternative: an ABM wherein simulated individuals are autonomous agents assigned with daily tasks that must be accomplished to the best of their abilities, depending on the context. These tasks are computed from TUS data and have several characteristics, such as a preferred period (the time frame during which the activity should preferably take place), duration range (minimum and maximum acceptable durations), frequency (number of repetitions during a day or a week). At each 1-minute timestep, simulated individuals replan their behavior by computing the priority of all their undone tasks and selecting the one with the highest priority. The priority computing takes numerous factors into account: the current task (agents prefer finish ongoing tasks rather than starting a new one); the other tasks they should be doing at the present time (i.e. all activities whose preferential period includes the current time step) because the more an agent is pressured by time, the less time it has for each tasks; the time remaining in the preferred period compared to the minimum duration of the task, etc. Exceptional events and the environment are considered in that process. For example, the impact of weather was computed for each type of individual based on TUS data, thanks to its "weather" column. Good weather increases the mean duration of leisure outside tasks and decreases the mean duration of inside leisure ones. Consequently, even if the list of tasks and their characteristics are identical, simulated individuals will not perform the same activities in the same sequence with the same duration on two consecutive days. This particularity ensures a first answer to L2: the activity variability comes from the ABM itself. The task generation process, detailed in [START_REF] Reynaud | Using Time Use Surveys in Multi Agent Based Simulations of Human Activity[END_REF], ensures coherence by extracting task sequences from real timetables. Variability is achieved by using aggregated data from individual types to select task time characteristics such as duration and preferred period. Minimum and maximum durations as well as preferred periods for each activity are computed such that they represent X% of the data, centered around the mean. For example, if X=50%, it means that, for a specific type of individual, for a specific activity, for a specific type of day, 50% of all repetitions of this activity in the data have a duration between the minimum and the maximum duration, centered on the mean duration. Thus, it is possible to increase or decrease the activity variability by increasing or decreasing X. If X=100%, simulated individuals are free to choose the duration of each activity if this duration exists in the data. As X approaches 0, the simulated individuals become more constrained to perform tasks closer to their mean duration. As an example, for the "work" activity of active women over 50 during the week, if X is 90%, the preferred period is from 6:00 AM to 7:50 PM, with a duration between 2 and 12 hours (representing 90% of activities in TUS data). If X is 50%, the preferred period is shortened to 7:40 AM to 4:50 PM, with a duration between 5 and 10 hours.

Regarding L4, some previous models (e.g., [START_REF] Yamaguchi | A stochastic model to predict occupants' activities at home for community-/urban-scale energy demand modelling[END_REF] worked toward the modelling of interactions among household members, but these interactions are often stereotyped (e.g., mandatory shared meals or no simultaneous use of the bathroom). In our model, we use the "who is present" column in TUS data to determine the activity's collectivity level: the percentage of times that this activity was carried out with other people. This allows for a priority bonus to be applied to other individuals when one is performing the activity, which increases activity variability. For instance, our simulations show that the "cooking" activity is performed with others about 60% of the time, while "housekeeping" is only done with others about 2% of the time. Some decisions are made at the household level. Washing clothes should be considered collectively on a weekly basis, as the number of washing machine cycles required depends on household size rather than individual usage. Another example is the cooking activity. Cooking for a meal taken together should be considered at a collective level since everyone does not only cook for themselves; if someone cooks, potentially everybody can eat.

Appliance use and energy consumption model

We now introduce how the model and data were used to improve the appliance use model. Allocating appliances to activities and setting the probabilities of use (PU) is challenging because of the L1 and L5 TUS limitations. While obvious associations can be made, for instance, "watching TV" that requires a TV, TVs can also be turned on much more often than just during the "watching TV" activity (e.g., when used for background visual and sound). Some activity-appliance associations are more probabilistic in nature (e.g., people do not use the vacuum cleaner for all their housekeeping activities, nor do they use it for the whole duration of the activity). Moreover, some real activities do not appear in the TUS at all because their timespan is lower than the 10 min threshold or are not considered to be a main activity. In SMACH, L1 is tackled by associating each task with one or several Appliance Use Models (AUM), that define the probability of a given appliance to be used during the task, and its operating mode. A task can therefore trigger the use of several appliances. To reflect the variety of real appliances and the way they are used, we introduce three types of AUM: Forced, Fractional and Cycle. In the Forced AUM, the appliance is used during the entirety of the task duration (e.g., "watching TV" leads to the TV being used for the entire duration of the viewing). In the Fractional AUM, the appliance works during part of the task and its use is scattered over the task duration (e.g., a vacuum cleaner whose use is disseminated during the housekeeping period). In the Cycle AUM, a cycle of the appliance is initiated, meaning that the appliance is used during a predetermined period that is not dependent on the duration of the task. This mode is applied to washing machines, dryers, dishwashers, and ovens. Some other appliances do not rely solely on human activity like heating or refrigerators and are not controlled by an AUM. Their thermal model have setpoint temperatures. Activity influences their operation, for instance through setpoint modifications for heating or refrigerator openings during meal preparation. The use of these AUM has the following advantages: -Seasonality: PUs can be changed based on the time of the day, type of day, or the season. For instance, dryers are documented to be much more used during winter than summer. Another example is the strong difference between Saturdays and Sundays, visible in the read load curve data used for the validation and presented below; -Variability: PUs enable different activities to have distinct load profiles, involving various appliances; -Readability/explainability: PUs are easy to understand and manipulate, as illustrated in the Use Case section.

To calibrate the appliance model PUs, we leveraged the CONSER study (EDF, 2016), which is based on a questionnaire completed by 4000 households regarding their electrical appliances and their appliance use habits (approximate duration and time of use within five daily periods). The resulting data are cross-referenced with technical information and research results on appliances power to estimate each appliance's yearly unit energy consumption [START_REF] Binet | Knowing residential electrical usage: a major challenge[END_REF]. We use these data to select the appliances that require explicit depiction based on their unit powers, annual unit energy and ownership rate. We grouped appliances that do not require an explicit model into composite appliances that represent sets of appliances used for similar activities like cooking, hygiene, or digital practices. Their energy usage is included in a baseline that varies with time. For example, the cooking appliances consist of six components: an electric oven, an electric cooking plate, a microwave oven, a coffee machine, a kettle, and the kitchen baseline. The calibration of the AUM was performed with help of the reference results of the CONSER study, which provides energy consumption targets in TWh for each category of appliances at the French population scale. The data used come from the updated and enriched 2019 version of this survey conducted in 2016, the results of which are presented in Figure 1. It appears that the targets in energy consumption cannot be satisfied if only the obvious association of activity and appliances are considered (e.g., "oven" only associated with "cooking"), even with a PU=1. It is therefore necessary to allocate AUM of some specific appliances to more activities than through a direct association. Examples of PUs for 3 appliances are shown in Table 1. At the beginning of each task, the selection of the devices that will be used is done by a random draw based on their PUs. PU values are first estimated through expert considerations (the use of a coffee machine is more probable during breakfast than during dinner), and then refined with the help of real hourly power demand data. Finally, an automatic calibration process is used to set the unit power of each appliance category based on the CONSER survey results. The parameters of the AUM are adapted to match the daily power reference data. For instance, in addition to the use of the CONSER results, the average cooking power at the population scale (29M households) was also adapted to fit the ADEME WattGo 2016 study (a study based on on-site measurements of appliance by appliance electricity consumption from 118 French households between 2014 and 2016): The validation of our model on activity and energy consumption is a continuous work as described in [START_REF] Schumann | Multi-Agent based simulation of human activity for building and urban scale assessment of residential load curves and energy use[END_REF]. A future publication will detail the validation including DHW and heating, while this paper focusses on specific electricity. The validation of aggregated power loads and energy consumptions was conducted by comparing model results with aggregated data from the "Panel Particuliers" (PP) panel of consumers. The data consist in French household power demand records at the timestep of 30 minutes, recorded for a maximum duration of 2 years. We used the answers of the participants about their home to generate a synthetic population representative of the panel (e.g., appliances, housing, inhabitants, habits such as holidays or weekend absences). We selected 300 households without electric heating nor electric DHW that exhibited power records of acceptable data quality and reliable questionnaires between March 2019 and February 2020 (pre-Covid). The 30min time-step averaged power demand for each month (i.e., the average of the 4 weeks of each month) were compared using a set of complementary metrics. The Mean Absolute Error (MAE) assessed the proximity between the model and the data; the Root Mean Square Error (RMSE) helped, along with the MAE, to identify the presence of extreme values. The Mean Absolute Percentage Error (MAPE) measured the percentage error of the forecast, while the Weighted Average Percentage Error (WAPE) was useful for identifying near-zero values. The Mean Directional Accuracy (MDA) was employed to compare forecast directions, and the Fréchet distance served as a measure of similarity. As an illustration the power demand for February 2020 is shown in Figure 3. The results, supported by the values of the RMSE in Table 2, demonstrate a proper dynamic of the aggregated power loads, including the minimums and maximums, as well as the ability to represent the distinct shapes of Wednesdays, Saturdays, and Sundays: We also validated the seasonality of power demand.

Figure 4 shows the absolute values and relative deviation in average monthly power demand over the year:

Use cases and illustrations

The model was used to quantify the impact of behavioral conservation measures for energy efficiency and peak shaving ("eco-behaviors") on the national load curve during the 2022-2023 winter energy crisis in France. We evaluated which behaviors (e.g., changes in showers, cooking, laundry) would be the most appropriate to 0% -2% -4% -11% -10% -8% -2% -2% -6% -3% 1% 3%

reduce the power demand during the peak hours defined by the TSO (8:00-13:00 and 18:00-20:00).

The first Use Case is an eco-behavior consisting in refraining from using cooking appliances during peak hours. As stressed by [START_REF] Shove | The dynamics of social practice[END_REF], practices constrain each other and shifting an activity will have an impact on the other activities. Families, particularly parents with young children, prefer not to postpone evening activities for too long and try to anticipate when they can [START_REF] Guibourdenche | Concerns and arrangements in contexts of domestic activity. Contribution to the design of energy-efficient and appropriable ubiquitous computing environments[END_REF]. Those diverse behaviors can be achieved with our activity model by displacing the preferential periods (PP) of cooking tasks outside of peak hours, with a maximum shift of 45 minutes to represent the limited possible delay in evening activities. This leads to a diversity of behaviors among agents, some choosing to anticipate their cooking tasks, others opting to postpone them, and some maintaining their regular schedule, rather than enforcing a uniform behavior on all agents.

Figure 5 shows the effect of such behavior on the activity rate and the average cooking appliances power demand. It is drawn from a 1000 dwellings simulation. The 34 possible tasks in the model were grouped in 8 categories.

Most of the cooking and eating tasks are delayed after 20:00 and few of them are done sooner. The released time is replaced mainly with leisure taks and idle (respectively an 8% and 9% rise at 19:00). "Idle" occurs when an agent does not have any specific task available. The power gain is substantial, reaching a maximum of 250W. At the beginning of a hygiene task, the occurrence of a shower is randomly drawn considering the day of the week and the number of showers already taken during the current day and since the beginning of the week. The ecobehavior consists in setting to zero the probability of a shower during peak hours. As there are often several hygiene tasks during the day, agents are expected to take advantage of off-peak hygiene tasks to take their showers. Figure 6 presents both the average hot water consumption and the average DHW power demand of a mean weekday with and without showers during peak hours. Hot water consumption is essentially transferred from peak hours to the evening. The reduction of power demand however is limited and occurs only in the morning, with a limited rebound effect at night. In France, a majority of electrical DHW are controlled to heat water only at night and in the beginning of the afternoon, when the power demand is low, which makes the relation between hygiene activities and power load very indirect.

Conclusion and perspectives

To overcome the limits related to the TUS and their use (L1-L5), we presented an Agent-Based Model combined with qualitative and quantitative studies. L3 is addressed by the population generator. L2 and L4 are addressed via the agent-based activity model and the inputs of qualitative studies which also help understanding L5. L1 is addressed via the presented Appliance Use Models based on calibrated Probabilities of Use. The model proposes a holistic approach for the representation of human activity and improves the consistency of simulated activity and energy consumption of households. This model has immediate practical applications and was used to quantify the impact of behavioral flexibility measures. Our work currently focuses on better understanding the cognitive and sociological macro-determinants of stability and variability of activity and how to integrate them in an ABM architecture. Future works will focus more on local scales such as neighborhoods or cities, by better considering the geographical, economic and sociodemographic factors of the studied territories. One other major challenge is the correct representation of the heating load curve at a national scale, considering heating practices and behaviors (including air renewal and air quality) such as those documented in the survey conducted in 2018 in France [START_REF] Laurent | The potential for lower indoor temperatures -A detailed survey of current heating behaviours in French households[END_REF].

Figure 1 :

 1 Figure 1: Reconstruction of specific electricity consumption in TWh from the CONSER study These targets were studied in the model using a synthetic population of 1000 households statistically representative of the French population, their homes and appliances, and the current electricity tariff structure.

Figure 2 :

 2 Figure 2: Power of oven and cooking plate on an average day at the French population scale Validation We now demonstrate how the model led to fitting results compared to real load curve data at an aggregated scale.The validation of our model on activity and energy consumption is a continuous work as described in[START_REF] Schumann | Multi-Agent based simulation of human activity for building and urban scale assessment of residential load curves and energy use[END_REF]. A future publication will detail the validation including DHW and heating, while this paper focusses on specific electricity. The validation of aggregated power loads and energy consumptions was conducted by comparing model results with aggregated data from the "Panel Particuliers" (PP) panel of consumers. The data consist in French household power demand records at the timestep of 30 minutes, recorded for a maximum duration of 2 years. We used the answers

Figure 3 :

 3 Figure 3: Average power (mean week Feb. 2020)Table 2: Monthly RMSE (in W) between Model and PP

Figure 4 :

 4 Figure 4: Average monthly power demand (W)Use cases and illustrationsThe model was used to quantify the impact of behavioral conservation measures for energy efficiency and peak shaving ("eco-behaviors") on the national load curve during the 2022-2023 winter energy crisis in France. We evaluated which behaviors (e.g., changes in showers, cooking, laundry) would be the most appropriate to

  . Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Model PP Power demand (W)

Figure 5 :

 5 Figure 5: Activity rate and cooking power load (red) without (top) and with (bottom) eco-behavior The second Use Case is a behavior aimed at avoiding taking showers and baths during peak hours to lower DHW power loads. In our model, showers and baths may be triggered during a hygiene task. Each simulated individual receives a fixed number of showers per week based on a reference survey (ADEME/COSTIC 2016). At the beginning of a hygiene task, the occurrence of a shower is randomly drawn considering the day of the week and the number of showers already taken during the current day and since the beginning of the week. The ecobehavior consists in setting to zero the probability of a shower during peak hours. As there are often several

Figure 6 :

 6 Figure 6: DHW power (left) and hot water consumption (right) with and without the "No shower" eco-behavior This Use Case illustrates how the model is able consider the complexity of the link between activity and appliance use (L1) and to represent plausible behavioral modifications and the interactions between activities (L2).

Table 1 :

 1 PUs for a selection of tasks and appliances

	Task	Microwave	TV	Computer
	Cooking	0.64	0.05	0.25
	Computer	0.02	0	1
	TV	0.02	1	0
	Reading	0.02	0	0.06
	Housekeeping	0.02	0.16	0.19
	Breakfast	0.01	0.05	0
	Meal	0.08	0.05	0.06
	Personal time	0.01	0.16	0.19

Table 2 :

 2 Monthly RMSE (in W) betweenModel and PP

	Jan.	37.2	May	45.6	Sep.	43.7
	Feb.	39.2	Jun.	44.1	Oct.	38.7
	Mar.	46.1	Jul.	32.5	Nov.	40.0
	Apr.	47.6	Aug.	33.2	Dec.	43.4