Louis Hawley
email: louis.hawley@usherbrooke.ca

Rémy Rahem
email: remy.rahem@usherbrooke.ca

Wael Suleiman
email: wael.suleiman@usherbrooke.ca

External Force Observer for Small and Medium-sized Humanoid Robots ⇤

Keywords: Force observer, External force, Small humanoid robots, Kalman filter, FSR, IMU

External force observer for humanoid robots has been widely studied in the literature. However, most of the proposed approaches generally rely on information from six-axis force/torque sensors, which the small or medium-sized humanoid robots usually do not have. As a result, those approaches cannot be applied to this category of humanoid robots, which is widely used nowadays in education or research.

In this paper, we propose a Kalman filter-based observer to estimate the three components of an external force applied in any direction and at an arbitrary point of the robot's structure. The observer is simple to implement and can easily run in real time using the embedded processor of a small or medium-sized humanoid robot, such as Nao or Darwin-OP. Moreover, the observer does not require any changes to the robot's hardware, as it only uses measurements from the available force-sensing resistors (FSR) inserted under the feet of the humanoid robot and from the robot's inertial measurement unit (IMU).

The proposed observer was extensively validated on a Nao humanoid robot in both cases of standing still or walking while an external force was applied to the robot. In the conducted experiments, the observer successfully estimated the external force within a reasonable margin of error. Moreover, the experimental data and the MATLAB and C++/ROS implementations of the proposed observer are available as an open source package.

Introduction

Having a biped structure and articulated arms makes humanoid robots ideal candidates to perform a variety of manipulation and transportation tasks. However, ⇤ A preliminary version of this paper has been presented at the 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2018) and published in Hawley et al (2018) [START_REF] Hawley | Kalman filter based observer for an external force applied to medium-sized humanoid robots[END_REF]. a https://goo.gl/VkhejY the manipulation or transportation of an object may create instability for the humanoid robot if no measurements are taken to consider the added force, which is generated by the transported object, and adjust the walking gait. Moreover, physical interaction with a human, such as collaboratively transporting an object, also generates interaction forces that should be taken into account to prevent instability issues. In those cases, it is primordial to accurately estimate the forces acting on the humanoid robot.

As an example, the transportation of a heavy object on a rolling cart, as depicted in [START_REF] Rioux | Humanoid navigation and heavy load transportation in a cluttered environment[END_REF][START_REF] Rioux | Autonomous SLAM based humanoid navigation in a cluttered environment while transporting a heavy load[END_REF], can be greatly improved if the external force generated by the cart and the transported object can be accurately measured or estimated. Moreover, for a cooperative transportation task of an object with two humanoid robots [START_REF] Rioux | Cooperative Vision-Based Object Transportation by Two Humanoid Robots in a Cluttered Environment[END_REF][START_REF] Hawley | Control framework for cooperative object transportation by two humanoid robots[END_REF] or with a humanoid robot and a human [START_REF] Oztop | Humanhumanoid interaction: Is a humanoid robot perceived as a human?[END_REF], it is crucial to measure the force exchanged through the transported object in order to maintain the equilibrium of the whole system.

! " In this paper, we consider the case of an external force applied to the humanoid robot as illustrated in Fig. 1.

F ext M c ẍc M c ÿc M c zc ⌧ y ⌧ x M c g

Relevant Works

There are multiple approaches to estimate external forces exerted on a humanoid robot. These approaches can be roughly separated into three main categories : October 26, 2019

External Force Observer for Small and Medium-sized Humanoid Robots 3 sensor-based approaches, model-based approaches and alternative methods such as machine learning-based methods. Each category is summarized with some examples in the following sub-sections.

Sensor-based Approach

The sensor-based approach directly measures the external force applied to a humanoid robot by using the robot's sensors. Such a method was used in [START_REF] Harada | ZMP Analysis for Arm / Leg Coordination[END_REF][START_REF] Harada | Pushing manipulation by humanoid considering two-kinds of ZMPs[END_REF], where a pushing manipulation task was performed by a HRP-2 robot. In that case, the external force applied to the humanoid robot can be directly measured at all time using the six-axis Force/Torque (F/T) sensors embedded in the robot's wrists. Then, by considering the external force, the authors were able to apply the required compensation on the torso position to ensure the stability of the robot while pushing the object.

A similar approach has also been proposed by the same authors to enable a humanoid robot to lift and transport a heavy object [START_REF] Harada | A humanoid robot carrying a heavy object[END_REF]. The sensor-based approach obviously provides great performance, since the force can be accurately measured at all times. However, F/T sensors are not available on all robots and it would be di cult to integrate them on smaller and cheaper robots due to the size and cost of such sensors.

Model-based Approach

We define model-based approach as the methods that estimate the external force acting on a humanoid robot by using a dynamic model of the robot and its actual state vector. For instance, the method presented in [START_REF] Kaneko | Disturbance observer that estimates external force acting on humanoid robots[END_REF], where the magnitude of an external force is estimated using a disturbance observer. The observer uses measurements from the robot's IMU and from F/T sensors located in the ankles to estimate the magnitude of a strong force, such as a kick to the robot's chest. In order to estimate the disturbance, the authors used a simplified dynamic model of the humanoid where the acceleration of its Center of Mass (CoM) is a function of the forces applied to the robot, which are: the gravity, the floor reaction force and the unknown external forces. Since the acceleration of the robot's CoM is known using IMU measurements and the floor reaction force can be directly measured with the F/T sensors, their results show that a strong impact force can be estimated somewhat accurately using that model. Another interesting method that can also estimate the external moment is proposed in [START_REF] Benallegue | Model-based external force/moment estimation for humanoid robots with no torque measurement[END_REF]. That method relies on a modelbased estimator to fuse the data of the robot's sensors (F/T sensors and IMU) and the whole body dynamics. An advantage of this approach over the previously presented sensor-based approach is that the external force/moment do not have to be applied at a particular point in order to be estimated. However, these methods still require F/T sensors.

A similar approach has been presented in [START_REF] Kaneko | Slip observer for walking on a low friction floor[END_REF] to enable a humanoid robot to walk on low friction floor. Using the linear inverted pendulum (LIP) model to represent the humanoid robot, the authors predicted the ground reaction force and compared it with the measurements from F/T sensors in the ankles of a HRP-2 robot. The di↵erence between the two values was then attributed to a slip force that can be easily derived. Although a slip force is also a strong force of short duration, which is very di↵erent from the force encountered in interaction tasks between several robots or a robot and a human, these works are interesting as they show that it is possible to get a good estimation of an external force by only using a limited number of sensors and a simple dynamic model of the robot. Such an approach would be interesting for medium-sized humanoid, such as Nao humanoid robot, as they usually have a small number of sensors and limited computational power. Nao [START_REF] Gouaillier | Mechatronic design of NAO humanoid[END_REF] is a medium-sized humanoid robot manufactured by SoftBank Robotics.

In [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF], the Nao robot's limbs are modeled as kinematic chains attached to the pelvis of the robot, and a relation between a force acting on each link of the robot and the joint torques is established, a method commonly used for serial manipulators. Since Nao's joints do not possess torque sensors nor do they provide a torque estimation, the authors had to define an actuator model to estimate the motor torque from the steady-state error. Then the problem is transformed into isolating the torque caused by gravity to obtain the portion of the torque that is the result of the external force. Although this approach can be quite powerful as the magnitude of the force along with the contact point can be found, there are multiple limitations in the proposed implementation. For instance, it uses the motors steady-state error to estimate the actuator torque. That means the proposed algorithm is only able to estimate big external forces.

Rotella et al. [START_REF] Rotella | Humanoid Momentum Estimation Using Sensed Contact Wrenches[END_REF] presented some formulations to estimate the CoM position of a robot and to correct inaccuracies in the robot links model. With regard to our work, the most interesting contribution is an external wrench estimator that uses a linear and angular momentum model of the robot instead of the simplified LIP model dynamic. However, the approach is limited to humanoids equipped with F/T sensors at the end-e↵ectors.

Another model-based method to estimate the magnitude of an external force applied to a humanoid robot is presented in [START_REF] Hawley | External Force Observer for Medium-sized Humanoid Robots[END_REF]. Its main advantage is that it does not require using F/T sensors but instead it uses measurements from the robot's force-sensing resistors (FSR) sensors. The approach has been successfully validated on a Nao robot and pointed out that an external force applied to the robot can be e ciently estimated with an acceptable precision. However, the main limitation of that method is that the formulations are only valid for an external force applied in the horizontal plane. That limitation has been addressed in [START_REF] Hawley | Kalman filter based observer for an external force applied to medium-sized humanoid robots[END_REF] to handle an external force applied in any direction.

Alternative Methods

Machine learning based approaches have also proven to be useful in estimating an external force applied to medium-sized/small humanoid robots.

In [START_REF] Berger | Inferring guidance information in cooperative human-robot tasks[END_REF], perturbations transmitted to a Nao robot are detected by comparing the actual sensor measurements with the predicted ones obtained with a probabilistic model. Using this method, the authors were able to successfully detect and provide an estimation of the perturbation magnitude. However, the external force that generates the perturbation has never been quantified since the algorithm can either provide the di↵erence between the sensors (IMU and FSR) measurements and the predicted values or the di↵erence between the expected footstep length and the equivalent footstep length. Although it is certainly possible to estimate the force from these results, such analysis has not been conducted.

In [START_REF] Carpentier | Center-of-Mass Estimation for a Polyarticulated System in Contact -A Spectral Approach[END_REF], the authors present a complimentary filter formulation to estimate the CoM of a robot or human. The proposed filter uses a spectral approach to analyze the di↵erent methods used to estimate the position of the CoM (i.e. motion capture, contact forces and moments). Then the authors propose to fuse the data by optimally using each sensor. Obviously, this approach requires a diverse set of sensors to span the whole frequency range, a condition that is often not satisfied in the case of small humanoid robots.

Contribution

The main contribution of this paper is the design of a model-based observer that is able to estimate the three components of an external force applied to a humanoid robot. The force is projected onto the CoM frame and therefore can be directly used, for instance, in a pattern generator. The observer design and implementation architecture are given in Section 5.

This observer was also validated on a Nao humanoid robot through several experimental scenarios. The results of these experiments are analyzed in Section 6.

ZMP with an external force

To study the impact of an external force on the Zero-Moment Point (ZMP) formulation [START_REF] Vukobratović | Zero-Moment Point -Thirty Five Years of Its Life[END_REF], we consider the two following cases:

An External Force Directly Applied to the Robot's CoM

This case study is illustrated in Fig. 2. The ZMP can be found by considering the forces acting on the simplified LIP model, which are the gravity, the inertial force and the external force. Let the ZMP on the horizontal ground be given by the following vector b :

F ext M c ẍc M c ÿc M c zc ⌧ y ⌧ x M c g
p =  p x p y (1)
where p x and p y are respectively the projections of the ZMP on the x and y axis.

To compute p, one can use the following formula:

p = N n ⇥ ⌧ o (f o |n) (2)
where the operators ⇥ and (.|.) design respectively the cross and scalar products, and

• N is a constant matrix N =  1 0 0 0 1 0 (3)
• the vector n is the normal vector on the horizontal ground

⇣ n = ⇥ 0 0 1 ⇤ T ⌘ . •
The vector f o is the result of the gravity, inertial and external force:

f o = M c g M c Ẍc + F ext (4)
where g denotes the gravity acceleration (g = gn), M c is the total mass of the humanoid robot, and Ẍc = [ẍ c ÿc zc ] T is the acceleration of the CoM.

F ext = [F x F y F z ]
T is the external force applied to the humanoid robot.

b As a general rule, bold parameters are vectors or matrices

The normal force is given by the following expression:

(f o |n) = f o n = M c g M c zc + F z (5)
• ⌧ o denotes the moment of the external force and the moment of the force f o about the origin of the fixed world frame. It is given by the following expression:

⌧ o =X c ⇥ f o (6)
where An external force applied to a point of the robot structure generates an additional external moment applied to the CoM of the LIP model as shown in Fig. 3. For instance, this force could be the result of the robot pushing, pulling or lifting an object or interacting/collaborating with a human.

X c = ⇥ x c y c z c ⇤ T is
Similarly to the previous case, the resultant force applied to the LIP model is:

f o = M c g M c Ẍc + F ext (8)
However, the moment ⌧ o in this case becomes:

⌧ o =X c ⇥ f o + ⌧ ext (9)
By applying the formula in (2), we obtain:

p = 2 4 p x p y 3 5 = 2 6 4 x c + Mc zc f o n ẍc zc f o n F x 1 f o n ⌧ ext y y c + Mc zc f o n ÿc zc f o n F y + 1 f o n ⌧ ext x 3 7 5 = 2 6 6 4
x c + Mc zc f o n ẍc zc f o n ⇣ F x + ⌧ ext y zc ⌘ y c + Mc zc f o n ÿc zc f o n ⇣ F y ⌧ ext x zc ⌘ 3 7 7 5 (10)
It is clear from [START_REF] Kaneko | Disturbance observer that estimates external force acting on humanoid robots[END_REF] that, without additional sensors such as a six-axis F/T sensor, the components of an external force and moment cannot be estimated. However, by defining the following virtual force:

F ext = 2 6 4
Fx Fy Fz

F x + ⌧ ext y zc F y ⌧ ext x zc F z 3 7 5 (11)
One can figure out that, from the ZMP point of view, the impact of an external force F ext applied to a point of the robot is the same as a virtual force F ext applied directly to the CoM of the LIP model. Therefore, we only consider the latter case in the sequel of the paper.

Measurements

Using the robot's available sensors (FSR, IMU and joint encoders), the ground reaction force, the ZMP, and the acceleration and position of the CoM can all be estimated. Those variables will play the role of output (observation) signals for the Kalman filter based observer. Note that, in this work, all measurements are unfiltered (raw) signals.

Ground Reaction Force

The ground reaction force can be directly estimated using the FSR sensors. In the case of Nao robot, for instance, each foot possesses 4 FSR sensors with a force range between 0 and 25N, and the ground reaction force f o n is given by:

f o n = 0 B B B B @ 4 X i=1 F R i | {z } right foot + 4 X i=1 F L i | {z } left foot 1 C C C C A (12)
where F ⇤ i is the force measured at the i th FSR.

ZMP

Recall that the ZMP and Center of Pressure (CoP) coincide when the robot is in a stable configuration [START_REF] Vukobratović | Zero-Moment Point -Thirty Five Years of Its Life[END_REF]. Thus, the ZMP can be computed as follows:

p x = x CoP = P 4 i=1 F R i x R i + P 4 i=1 F L i x L i |f o n | p y = y CoP = P 4 i=1 F R i y R i + P 4 i=1 F L i y L i |f o n | (13)
where f o n is defined in [START_REF] Kaneko | Slip observer for walking on a low friction floor[END_REF], and x ⇤ i and y ⇤ i are the x and y positions of the i th FSR in the foot frame. The acceleration of the CoM can be directly extracted from the IMU, which is a standard part of a humanoid robot sensors. As for Nao, the IMU provides measurements from a three-axis accelerometer and a two-axis gyroscope. We assume that the CoM coincides with the IMU. Therefore, the accelerometer data can be used to approximate the acceleration of the CoM.

CoM Acceleration

Fig. 4 presents a block diagram of the raw acceleration data processing. The projection into an inertial frame and the gravity compensation steps are implemented as presented in [START_REF] Bellaccini | Manual guidance of humanoid robots without force sensors: Preliminary experiments with NAO[END_REF].

CoM Position

The CoM position in the support foot frame can be easily obtained by a simple forward kinematics.

External Force Observer

The first component of the external force applied to the humanoid robot that can be directly estimated from the FSR information is F z using [START_REF] Hawley | Control framework for cooperative object transportation by two humanoid robots[END_REF]. Then the horizontal components F x and F y can be estimated using [START_REF] Harada | ZMP Analysis for Arm / Leg Coordination[END_REF].

Estimating F z

To estimate F z , Eq. ( 5) is discretized with a sampling time T to obtain the following linear dynamic system:

Z(k + 1) = A Z(k) + B u z (k) Y z (k) = C z Z(k) + v z (k) (14)
where:

• The state vector Z is given by:

Z(k) = ⇥ z c (kT ) żc (kT ) zc (kT ) F z (kT ) Ḟz (kT ) ⇤ T
• The input signal u z in given by:

u z (k) = ⇥ ... z c (kT ) Fz (kT ) ⇤ T
where ... z c (kT ) is the acceleration derivative, which is usually called the jerk. • The output signal Y z is given by:

Y z (k) = ⇥ z c (kT ) zc (kT ) f o n (kT ) + M c g ⇤ T
• The model constant matrices A, B and C z are given by:

A = 2 6 6 6 6 6 4
1 T T 2 2 0 0 0 1 T 0 0 0 0 1 0 0 0 0 0 1 T 0 0 0 0 1 3 7 7 7 7 7 5 , B = 2 6 6 6 6 6 4 T 3 6 0 T 2 2 0 T 0 0 T 2 2 0 T 3 7 7 7 7 7 5
C z = 2 4 1 0 0 0 0 0 0 1 0 0 0 0 M c 1 0 3 5 (15)
October 26, 2019

External Force Observer for Small and Medium-sized Humanoid Robots 11

• v z is the observation noise and considered as a Gaussian white noise with a covariance matrix R z . Thus, v z ⇠ N (0, R z ).

To formulate the estimation problem as a Kalman filter, we reformulate the dynamic system in ( 14) as follows:

Z(k + 1) = A Z(k) + ! z (k) Y z (k) = C z Z(k) + v z (k) ( 16
)
where ! z (k) = B u z (k) becomes the process noise. ! z is supposed to be a Gaussian white noise with a covariance Q, i.e. ! z ⇠ N (0, Q), independent of v z . The matrix Q is given by:

Q = B " 2 ... z c 0 0 2 Fz # B T (17)
where 2 ... z c and 2

Fz

are respectively the covariance of ... z c and Fz . By applying Kalman filter on ( 16), an estimation of the state vector is obtained:

Ẑ(k) = h ẑc (kT ) ẑc (kT ) ẑc (kT ) Fz (kT ) Fz (kT ) i T (18)

Estimating F x and F y

As above-mentioned F x and F y can be estimated from [START_REF] Harada | ZMP Analysis for Arm / Leg Coordination[END_REF]. However, because of the similarity between the two components of ZMP (p x and p y ), we only consider the estimation of F x as F y can be estimated in an analogous way. At first, p x in ( 7) is discretized with a sampling time T to obtain the following linear dynamic system:

X (k + 1) = A X (k) + ! x (k) Y x (k) = C x X (k) + v x (k) (19)
where:

• X (k), ! x (k), v x , A and Q are defined in the same way as in Section 5.1 by replacing F z by F x and z c by x c . • The output signal Y x is given by:

Y x (k) = ⇥ x c (kT ) ẍc (kT ) p x (kT ) ⇤ T
• The output matrix C x is given by: By applying Kalman filter on [START_REF] Vukobratović | Zero-Moment Point -Thirty Five Years of Its Life[END_REF], an estimation of the state vector X is obtained:

C x = 2 6 4 1 0 0 0 0 0 0 1 0 0 1 0 Mc ẑc(kT ) f o n (kT ) ẑc(kT ) f o n (kT )
X (k) = h xc (kT ) xc (kT ) xc (kT ) Fx (kT ) Fx (kT ) i T ( 20
)
The observer implementation architecture is summarized in Fig. 5.

IMU

Experimental Results

The observer has been experimentally validated on the Nao robot through four scenarios, and it has also been used to accomplish human-robot interaction tasks. For the reported experiments results, the following parameters were used: g = 9.81 m.s 2 , T = 16.7 ms (F = 60 Hz) and M c = 5.19 kg.

Regarding the observer parameters, for the experiments 1, 2 and 3, the best results have been obtained with:

2 ... x = 2 ... y = 2 ... z = 10 3 2 Fx = 2 Fy = 2 Fz = 10 3 R z = 2 4 0.01 0 0 0 1 0 0 0 1 3 5 , R x = R y = 2 4 0.01 0 0 0 1 0 0 0 0.01 3 5 (21)
While for the experiment 4, the following parameters have been used:

2 ... x = 2 ... y = 2 ... z = 10 2 2 Fx = 2 Fy = 2 Fz = 10 3 R z = 2 4 0.1 0 0 0 1 0 0 0 1 3 5 , R x = R y = 2 4 0.1 0 0 0 1 0 0 0 0.1 3 5 (22)
As we mentioned before, the experimental data as well as C++/ROS implementation and the MATLAB code that has been used to produce the results in this paper can be downloaded here c . These data and code could be helpful to the humanoid robotics community with the objective of benchmarking or comparing with other approaches.

Force Applied to the Robot Torso

In this experiment, a force in the x z plane is applied through a spring-based force meter as shown in Fig. 6(a). Since the torso frame is close to the CoM position of the robot, the magnitude of resulting moment applied to the LIP model is relatively small. Therefore, applying a force to the robot torso is practically equivalent to applying it to its CoM.

The F x and F y components of the force can be easily obtained by the following formula:

F x = kF ext k cos(✓) F z = kF ext k sin(✓) ( 23
)
where kF ext k is the force magnitude measured with the force meter, and ✓ is the angle between the force meter and the x axis, the positive direction of rotation is defined to be clockwise. The experiment consists of applying a force with a constant magnitude of approximately 2N in the x z plane but with di↵erent values of the angle ✓.

Fig. 6(b) shows the overall performance of the observer, it points out that the observer reasonably succeeds in estimating the three components of the external force. However, note that the main di culty in this experiment was keeping a force with constant magnitude by only visually monitoring the force indicator. Moreover, the angle (✓) was also visually monitored using a protractor.

We can also notice that the observer detects non-zero force along y axis, this is mainly due to the di culty of keeping the force meter in the vertical plane. The setup for the second and third scenarios are respectively presented in Fig. 7(a) and Fig. 8(a), and the attached mass m o = 2.5 pounds (1.13 kg). In both experiments, a force of around 11 N has been applied to the robot right arm, first along the y axis in Fig. 7(a) and then along the z axis in Fig. 8(a).

Applying a Force to a Point of the Robot Arm

By applying inverse dynamics to the kinematic chain from the point at which the force is applied to the robot's CoM, the wrench vectors in the second and third experiments as well as the virtual forces, as defined in [START_REF] Benallegue | Model-based external force/moment estimation for humanoid robots with no torque measurement[END_REF], are respectively: Those three experiments are very similar to the experiments conducted in [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF]. As such, they can be used as a benchmark to compare with their results. Fig. 7(b) and Fig. 8(b) show that the proposed observer succeeds in estimating the corresponding virtual external force since the estimated force is reasonably close to the expected one.

W 2 =  F ext 2 ⌧ ext
The authors in [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF] point out that the estimated force, by their method, has an important magnitude instead of a small one when the mass is removed. They argue that their main sources of error are the unmodeled static friction and the low-cost hardware of Nao robot, for instance, plastic joint gears. On the other hand, the estimated force by our observer after removing the mass has a small magnitude of around 0.5 N, which is an order of magnitude smaller than the results in [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF]. Furthermore, those results also show the robustness of our observer to the sources of error identified in [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF].

Walking with an External Force

The experimental setup used in this case study is given in Fig. 9(a). Initially, the robot is standing still and attached from the waist to a mass which is in full contact with the ground. The robot then starts walking forward. At some point, the cable (fishing line) becomes completely tensed and the mass is lifted o↵ the ground by the walking robot, as shown in Fig. 9(b). The lifted mass (m 0 ⇡ 300 g) generates a force of approximately 3N, which acts as an external force pulling the robot backwards. Note that when the rope is tensed, a small angle ↵ ⇡ 5 is formed between the cable and the x axis.

The experimental results of a forward walk with an external force are presented in Fig. 10. The estimated F x and F y components of the external force F ext are given in Fig. 10( caused by the elasticity of the cable and its friction with the experimental structure, which explains the results of the observer in the dashed-red encircled area in Fig. 10. Analyzing the estimated F x points out that the observer correctly estimated the pulling force that propagated to the robot through the cable. However, some oscillations can be observed, which are caused by the cable elasticity on one hand, and because of the mass acceleration on the other. In fact the external force is F x = m 0 (g + z0 ) ⇡ m 0 (g + ẍc ), where z0 is the object acceleration along z axis.

By analyzing the total normal force applied to the robot feet, we can observe that the force is varying roughly between 35 and 60 N. A deeper analysis revealed that some of the FSR have lost contact with the ground during the walking experiment. As a result, the observer estimated that an external force along the z axis (F z ) were applied.

Compliant Arm Control

As an application of our observer, we implemented a compliant controller of the robot arms as a function of the detected force. A snapshot of the experiment is given in Fig. 12.

However, during the experiment we observed that the observer performance degraded. The main problem is that the interaction between the robot and the human often caused a bad contact between the feet and the floor, which results in unreliable data from the FSR.

Thereby, to implement a compliance control of Nao arms, we combined the proposed force observer with a joint level compliance strategy. It is well known that it is possible to estimate the magnitude and direction of an external force applied to a manipulator if an accurate dynamic model of the robot is available and if the torques applied by the motors are known [START_REF] Magrini | Estimation of contact forces using a virtual force sensor[END_REF][START_REF] Magrini | Control of generalized contact motion and force in physical human-robot interaction[END_REF]. This approach can also be extended to humanoid robots [START_REF] Mattioli | Interaction Force Reconstruction for Humanoid Robots[END_REF], albeit it is more di cult since humanoid robots are floating-base robots.

First, the torque due to the robot dynamics must be isolated. To do so, it is possible to use the following inverse dynamic equation:

⌧ m = B(q) q + C(q, q) q + g(q) + J T (q) F ext , ⌧ + J T (q) F ext ( 25
)
where J is the Jacobian matrix of the considered chain, ⌧ m is the measured joint torques, B(q) is the inertia matrix, C(q, q) q includes the Coriolis and centrifugal forces, g(q) is the gravity term, F ext is the external force applied at the end-e↵ector, and q, q, q are respectively the position, velocity and acceleration of the joints. The external force can then be easily obtained by subtracting this torque from the measured one to get the external force applied to the humanoid arms:

F ext = J † T (q) (⌧ m ⌧ ) (26)
where J † is the pseudo-inverse of matrix J.

Referring to Nao robots o cial documentation [START_REF] Aldebaran | Nao o cial documentation[END_REF], the Nao motors are controlled through a PD controller, making it possible to determine the motor output torque with the following linear approximation:

⌧ i m = K m (q d q e ) (27)
where ⌧ i m is the torque for the motor i, (q d q e ) is the position error and K m is axis (F x ), we implemented the arm compliance as follows:

x h = x 0 h + 1 k F x (29)
where x h and x 0 h are respectively the desired position of the hand and its initial position along the x axis, k is a virtual spring constant and F x is the detected force along x axis. We chose a sti↵ness coe cient k = 50 Nm 1 . Fig. 14 shows the performance of the proposed observer and the compliance controller. As a second application, we have implemented an interactive walking based on the interaction between the robot and a human as shown in Fig. 15. In this scenario, the robot walks in the direction of the detected force (forward or backward) if the force magnitude exceeds a predefined threshold, and stops if the magnitude of detected force is less than the threshold.

2

Fig. 15. Interactive walking by applying a sagittal guidance force to the robot hands

Conclusion

In this paper, we introduced a Kalman filter based observer to estimate the three components of an external force applied to a small or medium-sized humanoid robot.

In contrast to conventional methods that are mainly based on the use of expensive 6-axis force/torque sensors, the proposed observer uses measurement from forcesensing resistors (FSR) located under the feet of the robot and the information from the robot's IMU sensor. This observer was extensively tested on a Nao robot and the results revealed that it overcomes the limitations of the related approaches in the literature, which are mainly caused by the low-cost hardware components of the robot. However, there is still room for improvement, for instance, the covariance matrices R and Q in the observer implementation have been experimentally identified, however, they can be estimated using some existing methods in the literature, such as [START_REF] Mehra | On the identification of variances and adaptive kalman filtering[END_REF]. Moreover, even though the objective of our observer is to estimate an equivalent force, called virtual force, that is applied to the CoM frame, it is possible to combine it with methods for identifying the contact point, such as [START_REF] Flacco | Contact detection and physical interaction for low cost personal robots[END_REF]. This would allow the observer to provide the external force components at the contact point and in the world frame by a simple inverse dynamics calculation.

Furthermore, our experiments revealed that the performance of the observer can degrade when one or more of the FSR lose contact with the ground. One of the reasons of this phenomenon is the rigid design of the foot sole. A possible solution could be to replace the rigid sole by a soft and compliant one and to embed the FSR inside the flexible sole.

Future work will focus on integrating the external force observer into the pattern generator module, and human-humanoid robot or multiple humanoid robots collaborative tasks.

Fig. 1 .

 1 Fig. 1. An external force applied to a Nao robot

Fig. 2 .

 2 Fig. 2. An external external force applied to the CoM of a Nao robot

2 .Fig. 3 .

 23 Fig. 3. An external moment is applied to the CoM of the LIP model as a result of an external force applied to a point of Nao robot. The accelerations are omitted for clarity

Fig. 4 .

 4 Fig. 4. Block diagram of the processing done on the raw acceleration data

0 3 7

 3 ẑc (kT ) and f o n (kT ) = M c g M c ẑc (kT ) + Fz (kT ) are computed using[START_REF] Carpentier | Center-of-Mass Estimation for a Polyarticulated System in Contact -A Spectral Approach[END_REF].

FxFig. 5 .

 5 Fig. 5. Block diagram of the external force observer F ext = [ Fx Fy Fz] T

c

  https://goo.gl/VkhejY Estimating the three components of F ext . The expected force is in green dashed line.

Fig. 6 . 1 :

 61 Fig. 6. Experiment 1: Applying a fixed-magnitude force in the x z plane.

Fig. 7 .Fig. 8 . 3 :

 783 Fig. 7. Experiment 2: Applying a force along y axis

Fig. 10 . 4 :

 104 Fig. 10. Experiment 4: Estimating the components of the external force F ext = [Fx Fy Fz] T , the expected force is in green dashed line.

Fig. 11 . 4 :

 114 Fig. 11. Experiment 4: The total normal force (f o n ) applied to the robot feet

1 2 Fig. 12 .

 1212 Fig. 12. Compliant control of the robot arms

Fig. 14 .

 14 Fig.14. Displacement of the arms and the estimated external force using the observer in Fig.13

6. 5 .

 5 Compliant Human-Robot Interaction

Louis Hawley, Rémy Rahem and Wael Suleiman

Louis Hawley, Rémy Rahem and Wael Suleiman ! "

Louis Hawley, Rémy Rahem and Wael Suleiman (a) Experimental setup

Acknowledgment

This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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p e t B S u f O Y Q f s N 4 + A X t y j 6 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 q z F 6 M H

p e t B S u f O Y Q f s N 4 + A X t y j 6 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 q z F 6 M H

p e t B S u f O Y Q f s N 4 + A X t y j 6 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 q z F 6 M H K 1 and K 2 are proportional gains, they can be chosen as constant parameters such as K 1 + K 2 = 1, or dynamically as follows:

where K a is the number of active FSR that are in contact with the ground, and K t is the total number of FSR, which is in the case of Nao robot equal to 8. It can be noticed that with this strategy, the observer based on inverse dynamics only reacts when at least one FSR loses contact with the ground.

The performance of the compliance controller was validated by a spring-based force sensor. By applying the proposed observer to estimate the force along the x


Timothy Boerst
email: tboerst@umich.edu

Meghan Shaughnessy

Connecting mathematical knowledge with engagement in mathematics teaching practices

Keywords:

Research suggests that mathematical knowledge is likely to influence how mathematics is taught. In turn, how mathematics is taught impacts students' opportunities to learn mathematics. We report on a study examining the connection between preservice teachers' mathematical knowledge and the nature of their eliciting and interpreting of student thinking. Our findings suggest that preservice teachers elicit and interpret student thinking with more emphasis on student understanding in situations in which they have strong mathematical knowledge of an algorithm used by the student compared to situations in which they have weaker mathematical knowledge about the algorithm used.

Mathematical knowledge, dispositions, and pedagogical skill

Mathematical knowledge and pedagogical skill are crucial to effective teaching. Studies show teachers with substantial mathematical knowledge [START_REF] Hill | Effects of teachers' mathematical knowledge for teaching on student achievement[END_REF] and command of core teaching practices like formative assessment [START_REF] Black | Assessment and classroom learning[END_REF]) have a greater impact on student learning of content. In the United States (US), mathematical knowledge and pedagogical skill are assets delineated in standards used to appraise both accomplished teachers (NBPTS, 2012) and beginning teachers (CCSSO, 2013). That mathematical knowledge and pedagogical skill are important in mathematics teaching is seldom questioned, but this does not mean that the field has determined the nature of the relationship between them. Mathematics teaching is a complex professional activity involving teachers in interaction with students, content, and context [START_REF] Cohen | Resources, instruction, and research[END_REF]. It is easy to imagine a multitude of ways in which mathematical knowledge and pedagogical skill are mutually influencing, if not mutually defining. For example, mathematical knowledge impacts the content and characteristics of questions that teachers ask during a lesson. Oriented in the other direction, pedagogical skill impacts the extent to which a teacher has access to student thinking that can be a substantial driver of a teachers' own mathematical development. This paper focuses on understanding the interrelations between mathematical knowledge and pedagogical skill empirically with respect to preservice teachers who are enrolled in a teacher preparation program.

Prior studies on teacher effectiveness connect teachers' mathematical knowledge with the achievement of their students. Often these studies examine the number and type of professional development experiences [START_REF] Blank | The effects of teacher professional development on gains in student achievement[END_REF] or teachers' performance on mathematics content tests. When mathematics content tests are used, they tend to focus on general mathematical knowledge or mathematical knowledge identified as central to mathematics teaching [START_REF] Hill | Effects of teachers' mathematical knowledge for teaching on student achievement[END_REF]. Such studies provide a connection between the mathematics that teachers know and the achievement of their students. What these studies do not explain is the way in which teaching is shaped by mathematical knowledge in ways that generate enhanced achievement for students. In a comparative study of Chinese and US mathematics teachers, [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF] advanced the notion that the way that teachers hold mathematical knowledge (e.g. the depth of understanding of foundational ideas interwoven with knowledge of how those ideas are shown across the breadth of mathematical topics) supports their engagement in teaching. Some examples of engagement in teaching are the questions teachers are able to ask students, the examples they create, and their analyses of student work. Among other mathematical foci, Ma explored differences in teachers' understanding of subtraction algorithms. Her findings illustrate the potential impact of the teachers' understanding and the ways in which different types of understanding might provide the basis for teaching that would support students' learning about subtraction. Needed are studies that can show connections between teachers' knowledge and the ways in which they engage in teaching practices. This would especially support teacher preparation, where resources are most plentiful, to scaffold improvement.

Since mathematical knowledge is likely to impact/influence mathematics teaching, it is important to understand how those resources impact teaching. To enhance the utility of what is learned, it will be important to focus studies on practices of teaching that are routine and crucial to students' learning of mathematics. Two such practices are eliciting student thinking and interpreting student thinking. Teachers use these practices to encourage students to share their thinking about specific academic content in order to evaluate student understanding, guide instructional decisions, and surface ideas that will benefit other students [START_REF] Teachingworks | High leverage teaching practices[END_REF]. Research on learning supports the use of teaching practices that enable teachers to know about and build on students' ideas. It also shows that teachers who know about their students' thinking and use that knowledge to influence their instruction have a significantly greater impact on student achievement [START_REF] Black | Assessment and classroom learning[END_REF]. Hence, understanding the connection between mathematical knowledge and preservice teachers' engagement in the teaching practices of eliciting and interpreting student thinking is likely to be use to the wider field.

In this context, this paper reports on research driven by a desire to better understand the ways in which mathematical knowledge influences preservice teachers' engagement in eliciting and interpreting student thinking. We studied these teaching practices of preservice teachers within a mathematical content strand that dominates the primary gradesnumber and operationzooming in on the operation of subtraction. Within subtraction, we focus on the use and understanding of different algorithms to solve problems, an area in which preservice teachers experience difficulty connecting their knowledge of traditional algorithms to alternative approaches [START_REF] Son | Moving beyond a traditional algorithm in whole number subtraction: Preservice teachers' responses to a student's invented strategy[END_REF]. Our research was guided by the question: What is the relationship between preservice teachers' mathematical knowledge of particular mathematical algorithms and the ways in which they approach eliciting and interpreting student thinking about those algorithms?

Using simulations to learn about preservice teachers' engagement in eliciting and interpreting student thinking

A practical problem that arises is how to study the ways in which preservice teachers' mathematical knowledge of particular mathematical algorithms impact their eliciting and interpreting of student thinking. We chose to design and use simulations in which preservice teachers engage in eliciting and interpreting a student's thinking about the use of a particular subtraction algorithm. Simulations are approximations of teaching that place authentic, practice-based demands on a participant while purposefully suspending or standardizing some elements of a teaching situation (Shaughnessy & Boerst, 2018). For each simulation, we developed a piece of student work and then designed an elaborate protocol detailing how a student would: complete the process shown in the work; reason about the process used; talk and act when questioned about the process. Those taking on the role of the student (teacher educators) were trained so that they could use information from the protocol as the basis for engaging in the situation and so that they could respond in ways that aligned with the profile of student thinking and acting.

Our decision to use simulations was driven by four factors. First, simulations of practice have proven to be usable in many professions to gauge clinical knowledge and skills (e.g. [START_REF] Boulet | The use of standardized patient assessments for certification and licensure decisions[END_REF]. Second, simulations provide reliable access to teaching situations that other methods, such as interviewing real children or interviewing in school contexts, could not. Third, simulations allow for the standardization of the assessment context [START_REF] Shaughnessy | Complementary assessments of preservice teachers' skill with eliciting student thinking[END_REF]. Fourth, our goal required a design that would purposefully put preservice teachers in situations where their mathematical knowledge was low. The use of simulations allowed us to accomplish this while preserving our own ethical commitments to educating children and consistently positioning preservice teachers to work productively with children.

Methods

We collected information about the mathematical knowledge of preservice teachers and used that information to assign them to three teaching simulations that each strategically positioned preservice teachers to elicit a student's thinking about a particular algorithm that had been used to compute an answer to a subtraction problem. Specifically, the simulations were designed to present opportunities to elicit a student's thinking about algorithms about which the preservice teachers had different degrees (known to the researchers) of mathematical knowledge. Twenty-four elementary preservice teachers from one teacher preparation program participated. Because our goal was not to bound our analysis to any point in time during a teacher education program, we purposefully recruited preservice teachers who were at different points in the program. Specifically, two were at the beginning of the teacher education program, eight were at the midpoint of the teacher education program, and 14 were at the end of the program. Next, we describe the pre-assessment, the assignment of preservice teachers to specific simulations, and the administration of the teaching simulations.

Pre-assessment of mathematical knowledge

We designed an assessment of mathematical knowledge inclusive of four subtraction algorithms that could be encountered in a subsequent teaching simulation. These algorithms, shown in Figure 1, include: A) the US traditional algorithm for subtraction; B) adding up subtraction; C) subtracting from the base; and D) expand and trade subtraction. Largely, these algorithms represent processes that are either directly taught through common US mathematics curriculum materials or are likely to surface in some form during mathematics work in US classrooms. One was deliberately picked to represent an algorithm that would likely be unknown to preservice teachers. This was done because of the potential for inaccuracy to influence preservice teachers' understanding of the different algorithms. For each example, preservice teachers were asked to determine accuracy of the answer, describe the steps that the student likely used to arrive at the answer, provide an explanation about the mathematical validity of the student's method, and apply the student's approach to another problem. The intent was to capture information that would allow us to categorize each preservice teacher's mathematical knowledge of each algorithm as "strong", "moderate", or "weak" relative to the mathematical knowledge shown across the algorithms. For each preservice teacher, the subtraction algorithm about which they demonstrated the most understanding was designated as "strong" and the subtraction algorithm about which they demonstrated the least understanding was designated as "weak. Those algorithms that were neither strong nor weak were designated as "moderate."

Assignment of preservice teachers to particular simulations

We used information from these pre-assessments to assign preservice teachers to three simulations. One simulation involved a "student" using an algorithm for which the preservice teacher was characterized as having strong mathematical knowledge (relative to the other algorithms). Another simulation involved a "student" using an algorithm for which the preservice teacher was characterized as having weak mathematical knowledge (relative to the other algorithms). An additional simulation involved a "student" using an algorithm for which the preservice teacher was characterized as having moderate mathematical knowledge (relative to the other algorithms).

Administration of the teaching simulations

All simulations were facilitated through the same sequence of stages. First, the preservice teacher prepared for an interaction with one simulated student (i.e., a teacher educator taking on the role of the student) by reviewing a specific piece of student work and considering what would be relevant to ask the student. For each simulation, they had 10 minutes for this preparation work. Second, the preservice teacher elicited and probed the simulated student's thinking to understand the steps she took, why she performed particular steps, and her understanding of the key mathematical ideas involved. Preservice teachers had five minutes to interact with the student. All preservice teachers finished their interactions within this time frame. All simulations were video recorded to support scoring and research purposes. Third, following the simulation, the preservice teacher responded to questions about their interpretation of the student's process and understanding and their prediction about the student's performance on a similar problem. Interviewers followed up on the preservice teachers' responses as needed.

Methods of analysis

We analyzed the simulation performances in which the preservice teachers elicited student thinking and the follow up interview in which the preservice teachers interpreted student thinking. Because the simulations made use of highly specified protocols for the student's processes, understandings, and ways of being, we were able to use checklists to track on the absence or presence of particular moves (Shaughnessy, & Boerst, 2018). Our checklists for the eliciting portion of the teaching simulations are based on an articulation of "high-quality" eliciting of student thinking. For example, high-quality eliciting student of thinking includes: eliciting the process/steps involved in using the algorithm and probing understanding of the algorithm and ideas underlying the algorithm. The checklist includes specific things that the preservice teacher might do in the context of each teaching simulation (e.g., probes why borrowing works) and specific responses that the student provides based on their preparation and training (e.g., "When you borrow, you're not changing the problem. You're just moving the same amount to a different place) when prompted by the preservice teacher. The practice of interpreting student thinking includes describing the components of the algorithm used by the student, making claims about the student's understanding of the algorithm, and substantiating claims by drawing on evidence from a larger body of data gathered from the eliciting. The checklist includes specific things that the preservice teacher might do in the context of each teaching simulation (e.g., describes the student's understanding of why borrowing works). For interpreting, we also cross-check any evidence that a preservice teacher cited with the information gathered during the interaction with the student to ensure that they had the evidence that they cited. For each situation, we further specified crucial "core" components of the process and understandings that were at the heart of the student's algorithm, ones that would be particularly important to attend to in the interpretation.

Each performance was appraised by two members of the research team. Disagreements were resolved through review of the data and remediating differences of interpretation by developing and refining a codebook.

Findings: Connecting the preservice teachers' eliciting with their mathematical knowledge and disposition

As noted previously, we analyzed the quality of each preservice teacher's eliciting performance in a context in which the preservice had "strong mathematical knowledge" of the subtraction algorithm versus their performance in which they had "weak mathematical knowledge of the subtraction algorithm. We next turn to findings related to their eliciting and interpreting of student thinking.

Eliciting practice

For each teacher, we compared their eliciting performances in the strong mathematical knowledge simulation context (SM) and the weak mathematical knowledge simulation contexts (WM). There were no differences across the two conditions with respect to the preservice teachers' eliciting of the student's procedural steps to solve the problem. However, preservice teachers probed for the students' understanding of the process more frequently in the SM simulation (M = .792, SD = .2518) than in the WM simulation (M = .313, SD = .3848), t(23) = 5.468, p < .001. In other words, when preservice teachers had stronger understanding of mathematics of the student's algorithm (relative to the second algorithm), they more frequently elicited the student's understanding about core ideas.

Interpreting practice

For each preservice teacher, we compared their interpreting performances in the SM simulation context and the LM simulation context. Preservice teachers did not differ in their skills in interpreting the procedural steps of the algorithm used by the student in the two contexts. Almost all of the preservice teachers named all of the student's steps in both contexts. However, the quality of their interpretation of the student's understanding of the process that was used was significantly higher in the SM simulation (M = .500, SD = .3612) than in the WM simulation (M = .188, SD = .3234), t(23) = 3.498, p < .002. In other words, preservice teachers were better at interpreting the students' understanding in situations in which they had strong mathematical knowledge of the algorithm compared to a situation in which their mathematical knowledge was comparatively weaker.

Implications and next steps

Preservice teachers demonstrated different eliciting and interpreting skills across two situationsone in which they had relatively high mathematical knowledge of a particular subtraction algorithm being used by a "student" and one in which they had relatively weak mathematical knowledge of a different subtraction algorithm. Preservice teachers focused their eliciting and interpreting practices more on student understanding when they had relatively strong mathematical knowledge. In some ways this finding is not surprising. Strong mathematical knowledge of an algorithm may enable preservice teachers to know what to ask about in terms of understanding. Further, in interpreting student thinking, mathematical knowledge of an algorithm may impact preservice teachers' abilities to identify the mathematical understanding that are worth naming. To be clear, in our work with preservice teachers we have seen cases where preservice teachers with strong mathematical knowledge of a situation did not elicit much thinking from the student, but instead made assumptions about the process and the students' understanding based on a combination of the student's written work and their own understanding of the algorithm. A contribution of the study was gain a better sense of the degree to which such engagement was common (or not) in our context. While this study does not provide insight into the reason(s) behind the differences in the practices, this outcome is useful because it shows concretely that differences in practice predicted by researchers like [START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF] who had theorized that greater mathematical knowledge would enable engagement in teaching practices that support student understanding of mathematics. Some elements of the preservice teachers' eliciting and interpreting were not significantly different across the two situations. Notably, there were not marked differences in the extent to which preservice teachers elicited procedural aspects of the student's algorithm. Likewise, there were not marked differences in preservice teachers' interpretations of procedural aspects across the different algorithms. When coupled with the previous analysis, this indicates a nuanced relationship between knowledge and teaching practice. The degree to which mathematical knowledge impacts the focus of teaching practices with respect to some facets of mathematics (understanding) more than others (procedural) is interesting. It is plausible that differences were not seen due to the fact that eliciting information about steps in a mathematical process, like an algorithm, may be more straight forward than eliciting information about understanding and thereby are not as heavily impacted by the degree of mathematical knowledge one has about an algorithm. The similarity of eliciting and interpreting with respect to the steps in an algorithm is notable for the questions it suggests. What enables preservice teachers to elicit and interpret student thinking about the procedural aspects of an algorithm irrespective of the depth of their mathematical knowledge about the algorithm? What compels preservice teachers to elicit student thinking about the procedural aspect of an algorithm and to attend enough to the procedural aspects to be able later to produce accurate evidence-based interpretations?

A clear limitation of the study findings presented here is consideration of other factors that might be contributing to the differences in eliciting and interpreting practices. For instance, a teacher's mathematical dispositions may shape teaching practices in ways that impact students' opportunities to learn mathematics [START_REF] Thompson | Teachers' beliefs and conceptions: A synthesis of research[END_REF][START_REF] Philipp | Mathematics teachers' beliefs and affect[END_REF]. As an example, in our own work with preservice teachers, we frequently hear differing, and often strongly expressed, dispositions toward particular computational algorithms. These dispositions influence and give structure to how teachers perceive students' engagement in mathematics. Studies have illustrated ways in which dispositions can impact the classroom practices of teachers (e.g., [START_REF] Fennema | A longitudinal study of learning to use children's thinking in mathematics instruction[END_REF]. Some of the connections between teaching practice and content knowledge that we have shared in this paper could be explained by the influence of mathematical disposition. For instance, it is just as plausible that a positive mathematical disposition may result in preservice teachers believing that it is worth asking questions about understanding because they personally know that the algorithm can be understood as it is that strong mathematical knowledge is the resource driving that approach to the practice. Given the potential for mathematical knowledge and mathematical dispositions to be mutually influencing (e.g. the more robust the mathematical knowledge of the teacher about an algorithm the more favorable the disposition to engage in and with the algorithm), in current work, we are teasing apart their influences with respect to engagement in teaching practices.

Figure 1 .

 1 Figure 1. Pre-assessment subtraction algorithm examples Preservice teachers were shown two examples of students' use of each of the subtraction algorithms, resulting in eight total examples to analyze. Half of the examples showed how the algorithms could result in correct answers and the other half of the examples showed how the algorithms could result in incorrect answers.This was done because of the potential for inaccuracy to influence preservice teachers' understanding of the different algorithms. For each example, preservice teachers were asked to determine accuracy of the answer, describe the steps that the student likely used to arrive at the answer, provide an explanation about the mathematical validity of the student's method, and apply the student's approach to another problem. The intent was to capture information that would allow us to categorize each preservice teacher's mathematical knowledge of each algorithm as "strong", "moderate", or "weak" relative to the mathematical knowledge shown across the algorithms. For each preservice teacher, the subtraction algorithm about which they demonstrated the most understanding was designated as "strong" and the subtraction algorithm about which they demonstrated the least understanding was designated as "weak. Those algorithms that were neither strong nor weak were designated as "moderate."

Acknowledgments

The research reported here was supported by the National Science Foundation under Award No. 1535389. Any opinions, findings, and recommendations expressed are those of the authors and do not reflect the views of the National Science Foundation. The authors acknowledge the contributions of Merrie Blunk, Rosalie DeFino, Susanna Owens Farmer, Erin Pfaff, and Xueying Ji Prawat to the design of the simulations, data collection, and data analysis.


Nils Buchholtz
email: f.buchholtz@ils.uio.no

Katrin Vorhölter
email: katrin.vorhoelter@uni-hamburg.de

Anna Orschulik
email: anna.orschulik@uni-hamburg.de

Nadine Krosanke
email: nadine.krosanke@uni-hamburg.de

Using mixed-assessments to evaluate opportunities to learn in mathematics teacher education

Keywords: Summative assessment, formative assessment, e-portfolio, opportunities to learn, mixed methods

Focus and rationale

Many teacher education programs require practical activities, like internships in schools or teaching experiments. Our task as teacher educators is to evaluate these activities and to assess performancerelated achievements, either for certification reasons or for program development. However, the assessment of the development of teaching competence of pre-service teachers in practical teaching activities is methodically challenging. In our study, we wanted to overcome current challenges with this issue by using an innovative methodological approach. In our evaluation study on mathematics pre-service teachers' opportunities to learn [START_REF] Buchholtz | Combining and integrating formative and summative assessment in mathematics teacher education[END_REF], a mixed-assessment was applied, consisting of a summative assessment (SA) of pre-service teachers' perceived study content and a formative assessment (FA) on situation-specific learning opportunities during internships in schools.

Theoretical background

In their review of research on the assessment of competencies in higher education, [START_REF] Blömeke | Beyond dichotomies: Competence viewed as a continuum[END_REF] model competence as a continuum that comprises dispositions, situation-specific skills, and performance. They also point to the challenges of an appropriate assessment: Pure cognitive-analytical approaches (like knowledge tests) lack sufficient validity because relevant parts, like the situational or contextual facets, can be underrepresented, and pure performance-based assessments might neglect the contribution of dispositional resources. Assuming that "the whole is greater than its parts" (p. 9), [START_REF] Kaiser | Professional competencies of (prospective) mathematics teachers: Cognitive versus situated approaches[END_REF] recommend working with a broader range of combined and situated assessment formats that are able to cover processes mediating the transformation of dispositions into performance. Other authors such as [START_REF] Carr | Tracking the development of learning dispositions[END_REF] also understand learning as a situated activity; these authors point to the situatedness of what they call learning dispositions. Accordingly, they favour formative assessment (FA) approaches such as interviews and self-reports like those that appear in portfolios when it comes to the assessment of performance-related competence. However, unlike in summative assessment (SA) forms, FA forms presuppose that we have to give up the idea of comparing the achievements of preservice teachers in a standardised way.

Methodology and corresponding research design

Against this background, two basic ideas of the mixed methods methodology can be transferred to the level of assessment [START_REF] Kelle | The combination of qualitative and quantitative research methods in mathematics education: A "mixed methods" study on the development of the professional knowledge of teachers[END_REF]. Using a combination of SA and FA can extend the scope of the assessment, i.e., both widen and deepen the assessment by gathering as much information as possible on multiple components of the assessed phenomenon or different, but closely related phenomena. The basic idea of an integration of SA and FA is to assess one phenomenon with different assessment forms, for example by using SA in a formative way for giving feedback, or integrating FA as a part of an overall SA [START_REF] Buchholtz | Combining and integrating formative and summative assessment in mathematics teacher education[END_REF].

Respectively, in our study, a panel-survey measuring the professional knowledge and learning opportunities of 187 pre-service teachers was combined with the analysis of 30 individual eportfolios from internships in order to holistically address both dispositional and situation-specific aspects of learning opportunities to develop teaching competence. In particular, the FA during the internship was composed of 18 predetermined tasks affording situation-specific skills. The preservice teachers had to address the tasks during their observations of other teachers or their own teaching in school, and to work on them as a written contribution to their personal e-portfolio. In favour of an increased interpretability of assessment results, a mutual complementation of the collective findings from the panel study with the findings from individual FA was sought (e.g. when identifying evidence for learning opportunities in FA). This approach might also partly overcome the challenge posed by the inability to standardize FA in teacher education.

Results

With SA data, we were able to determine comparatively that the analysed pre-service teachers had significantly higher perceptions of the number of learning opportunities regarding heterogeneity and research in mathematics education. Additionally, we used the data from the FA to identify preservice teachers' situation-specific skills and to link the individual learning opportunities of the preservice teachers with specific situational aspects of the school internships. With this combination, we were able to make detailed statements about the types of learning opportunities the pre-service teachers had, for example when dealing with errors in the classroom or differentiating instruction.