Théo Besson

Hugo Bouxom

Thibault Jaubert

Halo It's Meat! the Effect of the Vegetarian Label on Calorie Perception and Food Choices

Keywords: Health halo effect, calorie perception, food choices, vegetarian food

In the last few years, vegetarian products have become a widespread dietary option in food industry to the point that large retail chains such as McDonald's has offered a vegetarian burger to their customers (e.g., the Grand Veggie). In the present research, two studies investigated the influence of a vegetarian label on calorie perception, frequency consumption and food choices. In the first study (N = 211) participants were randomly assigned to an experimental design 2 (burger type: Grand Veggie vs Big Mac). In the second study (N = 915), participants were either exposed to a similar burger labeled vegetarian or a meat-based. Both studies show that vegetarian products are perceived as being less caloric than their non-vegetarian equivalents. However, participants do not report more intention to eat more vegetarian products and do not lead to differences in menu composition.

On October 10 th , 2017, the fast food brand McDonald's launched a new vegetarien burger in france (the Grand Veggie). The company hopes to attract the new vegetarian and flexitarian 1 market rising in France (34% in 2016France (34% in compared to 25% in 2015;;[START_REF] Worldpanel ; Khan | Licensing effect in consumer choice[END_REF]. This rapid spread of the vegetarian diet is probably due to media coverage flaunting its health benefits, as red and processed meats have been classified as potential carcinogens to humans (World Health Organization, 2018).

However, it is known that foods are classified dichotomously and are considered either healthy or unhealthy [START_REF] Chernev | The Dieter's Paradox[END_REF][START_REF] Oakes | Too good to be true: Dose insensitivity and stereotypical thinking of foods' capacity to promote weight gain[END_REF][START_REF] Raghunathan | The unhealthy = tasty intuition and its effects on taste inferences, enjoyment, and choice of food products[END_REF][START_REF] Rozin | Lay American conceptions of nutrition : Dose insensitivity, categorical thinking, contagion, and the monotonic mind we propose that two simplifying approaches to nutritional[END_REF] and that many studies have shown that food labels influence this classification [START_REF] Chernev | Calorie estimation biases in consumer choice[END_REF] through the health halo effect. Derived from the halo effect [START_REF] Thorndike | A constant error in psychological ratings[END_REF], the health halo effect occurs when individuals infer from a positive health attribute others attributes. For example, sweets labeled as "low-fat" are considered to be less caloric than their unlabeled counterparts [START_REF] Wansink | No Can "Low-Fat" Nutrition Labels Lead to Obesity[END_REF]. Similarly, products labeled "gluten-free" are perceived as healthier, containing fewer calories and being less processed (Prada, Godinho, Rodrigues, Lopes, Garrido, in press). This effect has also been found on labels that do not focus on nutritional aspects, particularly on Fair Trade (Schuldt, Muller & Schwartz, 2012) or organic products [START_REF] Ellison | Putting the organic label in context: Examining the interactions between the organic label, product type, and retail outlet[END_REF][START_REF] Lee | You taste what you see: Do organic labels bias taste perceptions?[END_REF]Schuldt & Schwartz, 2010). It can therefore be assumed that the vegetarian label induces the same type of effect.

These labelsthrough the health halo effectmay be one contributor of overweight. The number of obese people in the world almost tripled between 1975and 2016, from 9% in 2000to 13% in 2016(WHO, 2018)). It is known that this raise is due both to an increase in the number of calories ingested [START_REF] Cutler | Why have americans become more obese?[END_REF] and to a decrease in physical activity (WHO, 2018). Labels omnipresent in food choices might therefore lead to misjudgments leading to an overconsumption of food. For example, cookies (i.e., oreos) tend to be consumed more if they were labeled organic than if they were not [START_REF] Schuldt | The "organic" path to obesity? Organic claims influence calorie judgments and exercise recommendations[END_REF].

It has also been shown that the people most sensitive to this halo effect are those who have the most positive attitudes toward the label. Thus, the effect is more pronounced for participants with explicit attitudes or pro-environmental behaviors (e.g., [START_REF] Schuldt | The "organic" path to obesity? Organic claims influence calorie judgments and exercise recommendations[END_REF][START_REF] Sörqvist | The green halo: Mechanisms and limits of the eco-label effect[END_REF]. This leads us to believe that people with favorable attitudes and behaviors toward vegetarian food (i.e., flexitarians; vegetarians; vegans) will be more sensitive to the health halo effect.

As labels, brands influence food categorization [START_REF] Chernev | Calorie estimation biases in consumer choice[END_REF]. Sandwiches from a brand with a healthier brand image (i.e., Subway) were rated as having 35% fewer calories than those from the brand with an unhealthy brand image (i.e., McDonald's). On the other hand, participants chose drinks, side dishes and desserts with almost one and a half more calories when their main dish came from the brand considered healthy than when it was not [START_REF] Chandon | The biasing health halos of fast-food restaurant health claims: Lower calorie estimates and higher side-dish consumption intentions[END_REF]. Hence, studying food distributors in addition to fast food chains are a real health interest to the extent that they offer a diet that is usually rich in energy [START_REF] Prentice | Fast foods, energy density and obesity: A possible mechanistic link[END_REF].

We therefore hypothesized that the vegetarian label would negatively influence the caloric perception of foods. Thus, the Grand Veggie will be perceived as being less caloric than the Big Mac (i.e., a classic meat burger). Similarly, Grand Veggie will be perceived as being able to be consumed more frequently than Big Mac. Participants in the Grand Veggie condition will be more likely to say they want to accompany it with a soda and a ration of french fries than those in the Big Mac condition. Finally, we believe that the halo effect will be stronger among flexitarian, vegetarian and vegan participants.

Study 1: assessment of mcdonald's veggie burger

Method

Participants and design Two hundred and eleven participants were recruited online (79.15% women, 20.37% men, M Age = 26.3, SD Age = 11.1) and randomly assigned to one group (burger type: Grand Veggie vs Big Mac).

Participants were asked to rate the caloric content and its recommended frequency of consumption compared to other McDonald's burgers. Then, they evaluated the probability that they would take the burger in a menu with soda and fries. Finally, they answered socio-demographic and diet-related questions.

Estimating the number of calories in burgers Participants were shown a screen with the burger picture (i.e., Grand Veggie vs. Big Mac) and the list of ingredients used to make it. We made sure that the participants had read the information mentioned above by asking them « Did you read the information above correctly? ». On the next page, participants assessed the number of calories in the burger compared to another (Compared to other burgers available at McDonald's, do you think Grand Veggie [Big Mac] contains more or less calories?) Participants responded on a 7-points Likert scale (1 = Fewer calories; 7 = More calories).

Evaluation of the recommended frequency of consumption

Participants assessed the recommended frequency of consumption (Do you think you can eat Grand Veggie [Big Mac] more or less often than other burgers available at McDonald's) on a 7-points Likert scale (1 = Less often; 7 = More often).

Evaluation of the intention to order a menu Finally, participants evaluated their intention to order this burger in a menu (If you decided to order a Grand Veggie[Big Mac]), how much do you think you would accompany it with a menu including a soda and a serving of fries?). They responded on a 7-points Likert scale (1 = No intention; 7 = Strong intention).

Demographic and dietary questions

Participants indicated their gender, age and socio-professional category. They specified their knowledge of McDonald's and for those who answered yes, also their frequency of use of the brand (i.e., I never go there, Less than once a year, Once a year, Once every six months, Once a quarter, Once a month, Once a week, Several times a week). They then indicated if they had ever consumed the Grand Veggie and the Big Mac. Finally, they specified their diet (i.e., Omnivorous; Flexitarian; Vegetarian; Vegan). All participants except those who declared themselves omnivorous were then asked to rank from most important to least important the reasons why they had adopted such a diet (i.e., Health; Ecology; Animal Welfare; Humans (famine & work)) 2 .

Statistical analyzes

We started by creating a hierarchical multiple linear regression model. First, we included the experimental condition and all control variables (i.e., gender, age, frequency of visits to McDonald's, previous consumption of a Grand Veggie, previous consumption of a Big Mac, and diet plan) to evaluate the effect of the vegetarian label on calorie assessment (Model 1a). Then, we added the interaction of the experimental condition and the diet plan inside the model (Model 1b). The same procedure was repeated for the other two dependent variables (i.e., recommended consumption frequency, intention to take a menu, respectively named Model 2a, 2b, 3a and 3b). All analyses were performed on Jamovi 0.9.5.16.

Results

Effect on caloric estimation

Model 1a (see Table 1): A first linear regression model including experimental condition (-1 = Grand Veggie; 1 = Big Mac), gender (-1 = woman; 1 = man), age, frequency of visits to McDonald's (1 = I never go there; 2 = less than once a year; 3 = once a year; 4 = once every six months; 5 = once every quarter; 6 = once a month; 7 = once a week; 8 = several times a week), Grand Veggie consumption (-1 = no; 1 = yes), Big Mac consumption (-1 = no; 1 = yes) and the diet plan (1 = vegan; 2 = vegetarian; 3 = flexitarian; 4 = omnivorous) was carried out with the caloric estimation score as a dependent variable (general model: F(7,203) = 2.46, p = .019, R 2adj = 0.05). The analysis revealed a significant main effect of the experimental condition, b = 0.19, p = .007 and gender, b = -0.14, p = .049. It 2): we entered the interaction between diet plan and label condition (general model F(8,202) = 2.23, p = .026, R 2 adj = 0.08). The analysis revealed a significant effect of gender, b = -0.14, p = .048 and a tendential effect of Big Mac consumption, b = 0.12, p = .099. However, we found no effect of the label condition, b = -0.14, p = .725; diet plan, b = -0.07, p = .419; age, b = -0.07, p = .315; frequency of visits to McDonald's, b = 0.12, p = .122; Grand Veggie consumption, b = 0.02, p = .752 and no interaction effect between the experimental condition and diet plan, b = 0.09, p = .412.

Effect on the possible frequency of consumption

Model 2a (see Table 3): A first linear regression model including experimental condition (-1 = Grand Veggie; 1 = Big Mac), gender (-1 = woman; 4): we added the interaction between diet plan and label condition (general model F(8,202) = 3.14, p = .002, R 2adj = 0.11). The analysis reveals a significant effect of the experimental condition, b = -1.54, p < .001 and the interaction of the experimental condition with diet plan, b = 0.42, p < .001. There is also a tendential effect of animal consumption patterns, b = -0.14, p = .071 and gender, b = -0.12, p = .092. However, there is no effect of age, b = 0.01, p = . Effect on the intention to take a menu Model 3a (see Table 5): A first linear regression model including experimental condition (-1 = Grand Veggie; 1 = Big Mac), gender (-1 = woman; 1 = man), age, frequency of visits to McDonald's (1 = I never go there; 2 = less than once a year; 3 = once a year; 4 = once every six months; 5 = once every quarter; 6 = once a month; 7 = once a week; 8 = several times a week), Grand Veggie consumption (-1 = no; 1 = yes), Big Mac consumption (-1 = no; 1 = yes) and diet plan (1 = vegan; 2 = vegetarian;

Discussion of study 1

We hypothesized a health halo effect based on the vegetarian label: participants in the Grand Veggie condition rated it as less caloric than participants in the Big Mac condition. However, they do not believe that it can be consumed more frequently than Big Mac except when the moderating effect of animal consumption habits is taken into account. We can therefore assume that these habits are, as we have postulated, a moderator of the health halo effect based on vegetarian labels. People who consume the least amount of animal products would also be the ones who are most prone to the health halo effect based on the vegetarian label. However, the population of flexitarians, vegetarians and vegans is too small for us to have an acceptable power, so we cannot generalize the results obtained and further studies will be needed. In addition, it seems predictable that people who do not eat any meat at all (i.e., vegetarians and vegans) report being able to eat Grand Veggie more often than Big Mac. Contrary to our hypotheses, we found no effect of the label on the intention to have the burger in a menu comprising a soda and a portion of French fries. In this case, it can be assumed that the halo effect induced by the vegetarian labeltherefore on a single productdoes not encourage more side dishes, unlike a halo effect based on a healthy brand image (i.e., [START_REF] Chandon | The biasing health halos of fast-food restaurant health claims: Lower calorie estimates and higher side-dish consumption intentions[END_REF], as the effect would in this case occur for all products. A study comparing food choices made in an all-vegetarian restaurant with those made in a conventional restaurant would certainly answer this question.

Interestingly, we find that women attribute more calories to burgers than men. This is probably because they focus more on fat content than other macronutrients when assessing whether a product is healthy or not than men [START_REF] Oakes | Gender differences in perceptions of the healthiness of foods[END_REF]. However, this result must be taken with hindsight given that our sample is only 20% male.

Nevertheless, this research has several methodological limitations in addition to the sampling. On the one hand, Big Mac is bigger than the Grand Veggie and contains two meat-based steaks while the Grand Veggie has only one vegetable steak. This size difference may have had an effect on the caloric estimate that participants made of it.

On the other hand, the Grand Veggie received extensive media coverage during the dissemination of this research and many press articles highlighted that it contained far more calories than the Big Mac (i.e., 763kcal vs 503kcal) and we did not control the participant's knowledge on this subject, which can nevertheless constitute a significant bias leading to a lower effect. b = 0.04, p = .307; for the gender, b = 0.03, p = .409; for the time since the last meal, b = 0.05, p = .129 and for the interaction between the experimental condition and diet plan, b = 0.06, p = .289.

Discussion of study 2

Once again, our results confirm our main hypothesis: participants in the "Vege Whopper" condition rated their burger as less caloric than participants in the "New Whopper" condition. These results are consistent with the health halo effect literature, allowing us to conclude that a health halo effect is based on the vegetarian label. However, this effect does not seem to be moderated by the consumption of animal products. Moreover, contrary to what we had assumed and in line with the results obtained in the first study, participants in Vege Whopper condition did not order more caloric accompaniments than participants in New Whopper condition. This leads us to believe that the label has an effect on the food concerned solely, in this case the burger, and not on all the foods constituting a meal.

This time again, the study presents several limitations, particularly concerning the recruitment of participants. The number of flexitarian, vegetarian and vegan participants is far too low compared to the number of omnivorous participants for us limiting the statistical power. Further research will be needed to conclude with more precision on the hypotheses involving participants who eat little or no animal products.

General discussion

In these two studies, we were able to highlight the existence of a health halo effect based on the vegetarian label. This discovery joins the long list of other labels with the same effects such as low-fat [START_REF] Wansink | No Can "Low-Fat" Nutrition Labels Lead to Obesity[END_REF], organic [START_REF] Besson | The calories underestimation of "organic" food: Exploring the impact of implicit evaluations[END_REF], gluten-free [START_REF] Prada | The impact of a gluten-free claim on the perceived healthfulness, calories, level of processing and expected taste of food products[END_REF] or fair-trade labels [START_REF] Schuldt | The "Fair Trade" Effect: Health halos from social ethics claims[END_REF]. The vegetarian label (like the others) seems to induce a lower caloric evaluation of food which could be problematic insofar as people on a diet could misunderstand when making decisions leading to eat food as much or even more caloric than if they had eaten meat-based food. However, since processed meats and red meat are involved in a number of diseases (World Health Organization, 2018), this vegetarian halo effect could lead to a decrease in the amount of meat consumed per year. Therefore, the implications of this effect can be both negative and positive. We can only encourage governments to make a nutritional score label mandatory, which makes it possible to provide global nutritional information when making purchasing decisions and encourage manufacturers and restaurateurs to change the composition of their products (see [START_REF] Hawkes | Smart food policies for obesity prevention[END_REF] for a review). However, it would seem that the health halo effect only concerns the labeled product and that it does not spread to the entire brand image and other products on sale. Similarly, it does not seem to generate a licensing effect in the construction of the menu to the extent that we did not observe any difference between the participants in vegetarian and meat conditions. This effect is the fact that a person who performs a virtuous action feels free to act in a less virtuous way afterward [START_REF] Worldpanel ; Khan | Licensing effect in consumer choice[END_REF]. This seems consistent with the previous literature which shows that for this effect to occur, it is essential that people intentionally choose the so-called virtuous food [START_REF] Prada | Deliberate choices or strong motives: Exploring the mechanisms underlying the bias of organic claims on leniency judgments[END_REF]. The health halo effect does not therefore seem to spontaneously induce this licensing effect. Further research will obviously be needed to better understand the effect of labels, and in particular the vegetarian label, on actual consumption. In addition, although diets containing few or no animal products do not appear to modulate the effect of vegetarian health halo in the case of burger marketed by an international fast food chain, it may be interesting to observe whether the situation is maintained in the case of vegetarian products (e.g., falafels) and in the case of products marketed by independent restaurants or with a healthier image (e.g., Prêt a manger). Finally, future studies will need to be more vigilant about the composition of their samples, not only by ensuring that there are more flexitarian, vegetarian and vegan participants, but also by having a better male/female distribution since men think more that meat is essential for good nutrition to the extent that the more they eat meat the less they display consideration of meat consumption on animal welfare (see Ruby 2012 for a review).

Notes

1. an individual who is neither vegetarian nor vegan but tends to reduce his consumption of animal protein. 2. We were unable to use this information in our analyses because people with specific diets represented only 19.9% of the sample. 3. 12 participants did not answer on their frequency of visit to Burger King.

1

  = man), age, frequency of visits to McDonald's (1 = I never go there; 2 = less than once a year; 3 = once a year; 4 = once every six months; 5 = once every quarter; 6 = once a month; 7 = once a week; 8 = several times a week), Grand Veggie consumption (-1 = no; 1 = yes), Big Mac consumption (-1 = no; 1 = yes) and diet(1 = vegan; 2 = vegetarian; 3 = flexitarian; 4 = omnivorous) was carried out with the caloric estimation score as a dependent variable (general model: F(7,203) = 1.27, p = .27, R 2adj = 0.04). The analysis revealed a significant effect of Big Mac consumption, b = 0.15, p = .036. However, we find no significant effects of the experimental condition, b = 0.02, p = .746; of the type, b = -0.11, p = .11; of the diet plan, b = -0.10, p = .228, age, b = 0.01, p = .947; frequency of visit to Mc Donald's, b = 0.01, p = .923; Grand Veggie consumption, b = 0.08, p = .295. Model 2b (see Table

  870; frequency of visit to Mc Donald's, b = -0.03, p = .711; Grand Veggie consumption, b = 0.04, p = .584 and Big Mac consumption, b = 0.10, p = .155.

3

  = flexitarian; 4 = omnivorous) was carried out with the caloric estimation score as a dependent variable (general model: F(7,203) = 10.8, p < .001, R2adj = 0.247). The analysis revealed a significant effect of age, b = -0.40, p < .001 and diet plan, b = 0.29, p < .001. However, no effects were found for the experimental condition, b = 0.07, p = .247; sex, b = 0.08, p = .223, frequency of visit to Mc Donald's, b = 0.09, p = .196, because of having already Grand Veggie consumption, b = 0.04, p = .502 and because of Big Mac consumption, b = 0.02, p = .745. Model 3b (see Table 6): which adds to the first the interaction of the experimental condition with diet plan (general model F(8,202) = 9.45, p < .001, R2adj = 0.27). The analysis revealed a significant effect of age, b = -0.40, p < .001 and diet plan, b = 0.29, p < .001. However, we find no effect of the experimental condition, b = 0.14, p = .697; of the interaction of the experimental condition with meat consumption patterns, b = -0.02, p = .843; of the gender, b = 0.08, p = .224; the frequency of visits to Mc Donald's, b = 0.09, p = .192; Grand Veggie consumption, b = 0.05, p = .49 and Big Mac consumption, b = 0.02, p = .722.

Table 1 .

 1 Study 1 -Model 1a.

	Model Coefficients
							95% Confidence
							Interval
						Stand.
	Predictor	Estimate	SE	t	p	Estimate	Lower	Upper
	Intercept	4.31812 0.54570 7.913 < .001
	Experimental condition	0.24607 0.09000 2.734 0.007	0.1872	0.05219 0.3221
	Gender	-0.22107 0.11145 -1.984 0.049	-0.1367	-0.27268 -8.17e-4
	Age	-0.00836 0.00814 -1.027 0.306	-0.0702	-0.20504 0.0646
	Frequency of visits to	0.10439 0.06276 1.663 0.098	0.1311	-0.02431 0.2865
	McDonald's
	Grand Veggie Consomption	0.06220 0.14435 0.431 0.667	0.0311	-0.11131 0.1736
	Big Mac Consomption	0.19043 0.10359 1.838 0.067	0.1315	-0.00954 0.2724
	Diet Plan	-0.10937 0.15700 -0.697 0.487	-0.0556	-0.21309 0.1018

revealed a main trend effect of the frequency of visits to McDonald's, b = 0.13, p = .098 and Big Mac consumption, b = 0.13, p = .067. Finally, no effects were found for age, b = -0.07, p = .306; Grand Veggie consumption, b = 0.03, p = .667 and for diet plan, b = -0.06, p = .487.

Model 1b (see Table

Table 3 .

 3 Study 1 -Model 2a.

	Model Coefficients
							95% Confidence
							Interval
	Predictor	Estimate	SE	t	p	Stand. Estimate Lower Upper
	Intercept	4.17467 0.7045 5.9261 < .001
	Experimental condition	0.03770 0.1162 0.3245 0.746	0.02264	-0.1150 0.1602
	Gender	-0.23089 0.1439 -1.6048 0.110	-0.11279	-0.2514 0.0258
	Age	7.06e-4 0.0105 0.0672 0.947	0.00468	-0.1328 0.1421
	Frequency of visits to McDonald's 0.00784 0.0810 0.0968 0.923	0.00778	-0.1507 0.1662
	Grand Veggie Consomption	0.19583 0.1863 1.0509 0.295	0.07740	-0.0678 0.2226
	Big Mac Consomption	0.28239 0.1337 2.1117 0.036	0.15393	0.0102 0.2977
	Diet Plan	-0.24481 0.2027 -1.2079 0.228	-0.09834	-0.2589 0.0622

Table 2 .

 2 Study 1 -Model 1b.

	Model Coefficients
							95% Confidence
							Interval
						Stand.
	Predictor	Estimate	SE	t	p	Estimate	Lower	Upper
	Intercept	4.41372 0.55838 7.904 < .001
	Experimental condition	-0.18872 0.53659 -0.352 0.725	-0.1435	-0.9483 0.53774
	Gender	-0.22185 0.11154 -1.989 0.048	-0.1372	-0.2733 -0.01603
	Age	-0.00821 0.00815 -1.006 0.315	-0.0689	-0.2039 0.00989
	Frequency of visits to	0.09819 0.06326 1.552 0.122	0.1233	-0.0333 0.11157
	McDonald's
	Grand Veggie Consomption	0.04612 0.14578 0.316 0.752	0.0231	-0.1208 0.23025
	Big Mac Consomption	0.17460 0.10545 1.656 0.099	0.1205	-0.0230 0.19457
	Diet Plan	-0.12856 0.15885 -0.809 0.419	-0.0654	-0.2247 0.14045
	Experimental condition*Diet	0.11661 0.14188 0.822 0.412	0.0887	-0.1241 0.24518
	Plan

Table 4 .

 4 Study 1 -Model 2b.

	Model Coefficients
							95% Confidence
							Interval
	Predictor	Estimate	SE	t	p	Stand. Estimate Lower Upper
	Intercept	4.74609 0.6958 6.821 < .001
	Experimental condition	-2.56113 0.6686 -3.831 < .001	-1.5384	-2.3303 -0.6071
	Gender	-0.23555 0.1390 -1.695 0.092	-0.1151	-0.2489 0.2552
	Age	0.00166 0.0102 0.163 0.870	0.0110	-0.1218 0.0215
	Frequency of visits to McDonald's -0.02923 0.0788 -0.371 0.711	-0.0290	-0.1832 0.0499
	Grand Veggie Consomption	0.09970 0.1816 0.549 0.584	0.0394	-0.1022 0.2496
	Big Mac Consomption	0.18772 0.1314 1.429 0.155	0.1023	-0.0389 0.1795
	Diet Plan	-0.35954 0.1979 -1.816 0.071	-0.1444	-0.3012 0.0185
	Experimental condition*Diet Plan 0.69703 0.1768 3.943 < .001	0.4187	0.2093 0.5108

Table 5 .

 5 Study 1 -Model 3a.

	Model Coefficients
							95% Confidence
							Interva
	Predictor	Estimate	SE	t	p	Stand. Estimate Lower	Upper
	Intercept	4.1319 0.6919 5.972 < .001
	Experimental condition	0.1324 0.1141 1.161 0.247	0.0706	-0.0493	0.191
	Gender	0.1726 0.1413 1.221 0.223	0.0748	-0.0460	0.196
	Age	-0.0678 0.0103 -6.565 < .001	-0.3989	-0.5186 -0.279
	Frequency of visits to McDonald's 0.1033 0.0796 1.298 0.196	0.0909	-0.0472	0.229
	Grand Veggie Consomption	0.1231 0.1830 0.673 0.502	0.0432	-0.0834	0.170
	Big Mac Consomption	0.0428 0.1313 0.326 0.745	0.0207	-0.1046	0.146
	Diet Plan	0.8033 0.1991 4.036 < .001	0.2864	0.1465	0.426

Table 6 .

 6 Study 1 -Model 3b.

	Model Coefficients
							95% Confidence
							Interval
	Predictor	Estimate	SE	t	p	Stand. Estimate Lower	Upper
	Intercept	4.1026 0.7091 5.786 < .001
	Experimental condition	0.2655 0.6814 0.390 0.697	0.1415	-0.5747 0.6976
	Gender	0.1728 0.1416 1.220 0.224	0.0749	-0.0462 2.6602
	Age	-0.0678 0.0104 -6.552 < .001	-0.3991	-0.5193 -0.0417
	Frequency of visits to McDonald's 0.1052 0.0803 1.309 0.192	0.0926	-0.0469 0.0924
	Grand Veggie Consomption	0.1280 0.1851 0.692 0.490	0.0449	-0.0831 0.2385
	Big Mac Consomption	0.0477 0.1339 0.356 0.722	0.0231	-0.1047 0.1111
	Diet Plan	0.8092 0.2017 4.012 < .001	0.2885	0.1467 0.6434
	Experimental condition*Diet Plan -0.0357 0.1802 -0.198 0.843	-0.0190	-0.2084 0.1385

Study 2: experimental replication

We conducted a second experiment to address the limitations of the first. On one hand, we created a bogus burger presentation ad that did not exist at Burger King (i.e., New Whopper and Vege Whopper) and on the other hand, we tried to make it more ecological by fitting into the universe of a fast-food brand (i.e., Burger King), by reproducing the food selection flow of the control terminals and by using traditionally food sold at Burger King. Finally, we launched this study on the Internet by diversifying the means of dissemination as much as possible in order to have access to the largest possible population.

For this second study, we adjusted our hypotheses and proposed that the vegetarian label would negatively influence the caloric perception of foods. Thus, the Vege Whopper will be perceived as being less caloric than the New Whopper. Participants in the Vege Whopper condition will order more caloric foods than participants in the New Whopper condition. Finally, we believe that the halo effect will be all the more pronounced among flexitarian, vegetarian and vegan participants.

Method

Participants and design

Nine hundred and fifteen participants were recruited online (92.2% women, 6.8% men, M Age = 23.5, SD Age = 65.0) and randomly assigned to one group (burger type: Vegetarian vs Meat) and were asked to rate the calorie content compared to a classic Burger (i.e., the Whopper). Then, they choose the accompaniment (i.e., drink, side dish, dessert) they wanted to add for their meal and the menu size (i.e., Normal vs Maxi). Finally, they answered sociodemographic questions.

Estimating the number of calories in burgers

Participants read a text asking them to visualize themselves ordering from a fast food chain: "Imagine that you are at Burger King (a fast food chain) and that you order your meal on a terminal". In Vegetarian condition, participants could read: "You have already chosen your sandwich: Vege Whopper which is Burger King's new sandwich. Ingredients: vegetarian steak, cheese, pickles, salad, ketchup, onions, tomatoes, sesame seed bread, mayonnaise sauce ". In the other condition (meat condition), participants could read this message: "You have already chosen your sandwich: The New Whopper which is Burger King's new sandwich. Ingredients: beef steak, cheese, pickles, salad, ketchup, onions, tomatoes, sesame seed bread, mayonnaise sauce" (Appendix 1). We made sure that the participants had read the information mentioned by asking them "Did you read the information above?".

On the next page, participants estimated the calories contained in the burger compared to a traditional one in Burger King (Knowing that the Whopper (classic burger) contains 660 Kcal, do you think that the [New] Vege Whopper is more or less caloric than the Whopper?) Participants responded on a 7-point Likert scale (1 = Much less caloric; 7 = Much more caloric). We then asked them to rate their attractiveness for the burger (How appetizing do you think this burger is?). They answered on a 7-point Likert scale (1 = Not at all appetizing; 7 = Very appetizing).

Choice of complementary foods and menu

Following this first block, participants selected a number of accompaniments. A first page invited them to select a drink (e.g., Coca Cola; Volvic Juicy Strawberry; Nothing; Appendix 2), a side dish (e.g., Chili Cheese Nuggets filled with cheese; Fries; Garden Side Salad; Appendix 2) and a dessert (e.g., Donut; Waffle and vanilla ice cream; Muffin; Appendix 2). We took back the food offered at Burger King as well as the brand's visuals. They were presented on the screen as if they were on a control terminal in a restaurant. Finally, a last page asked participants if they would like to order a Medium menu (portions presented: 33cl drink, 114g side dish) or King Size (50cl drink and 160g side dish).

Calorie's estimations (i.e., per 100g) were based from the French and Swiss websites brands for each product

Behavioral intent

We asked participants to indicate their intention to consume the selected meal on a 5-point scale (How likely (out of 5) would you be to order this meal as part of a real order?).

Demographic and dietary questions

Participants indicated their gender, age and socio-professional category and the time elapsed since their last meal. They then specified diet plan (i.e., I eat animal flesh and animal products; I avoid the consumption of animal flesh, but this sometimes happens to me; I do not eat animal flesh, I do not eat animal flesh or animal products). We asked them if they knew the Burger King fast food chain and how much they frequented the store (i.e., I never go there, Less than once a year, Once a year, Once every six months, Once a quarter, Once a month, Once a week, Several times a week). Participants mentioned whether they had ever participated in similar studies and specified their content. Finally, they were invited to give their comments on the study.

Statistical analyses

We started by creating a hierarchical multiple linear regression model. First, we included the experimental condition and all controlled variables (i.e., finding the burger appetizing; the probability of actually ordering the burger; gender; age; the number of hours since the last meal; frequency of visits to the burger king 3 and meat consumption habits to evaluate the effect of the vegetarian label on calorie assessment (Model 1a). Then, we added the interaction of the experimental condition and the diet plan inside the model (Model 1b). The same procedure was repeated for the number of calories ordered (Model 2a and 2b). All analyses were performed on Jamovi 0.9.5.16.

Results

Effect on caloric estimation

Model 1a (see Table 7): A first linear regression model including the experimental condition (-1 = Vege Whopper; 1 = New Whopper) finding the burger appetizing; the probability of actually ordering this meal; gender (-1 = Female; 0 = Other; 1 = Male); age; the number of hours since the last meal; the frequency of visits to Burger King (1 = I never go there; 2 = Less than once a year; 3 = Once a year; 4 = Once every six months; 5 = Once every quarter; 6 = Once a month; 7 = Once a week 8 = Several times a week) and diet plan (1 = vegan; 2 = vegetarian; 3 = flexitarian; 4 = omnivorous) was achieved with the caloric estimation score as a dependent variable (general model: F(8,894) = 12.3, p < .001, R 2 adj = 0.09). The analysis revealed a main effect of the experimental condition, b = 0.28, p < .001 and age, b = -0.07, p = .033. There is also a tendential effect of 8): Which adds to the first the interaction of the experimental condition with diet plan (general model F(9,893) = 10.9, p < .001, R 2 adj = 0.09). The analysis revealed a significant effect of age, b = -0.07, p = .035 as well as tendential effects of the gender, b = -0.06, p = .098 and frequency of visit to Burger King, b = -0.06, p = .060. However, no effect was found of the experimental condition, b = 0.18, p = .362; due to finding the burger appetizing, b = 0.02, p = .472; time since the last meal, b = -0.01, p = .641; diet plan, b = 0.02, p = .536, the probability of actually ordering this meal, b = -0.03, p = .404 and the interaction between the experimental condition and diet plan, b = 0.03, p = .582.

Effect on the number of calories ordered

Model 2a (see Table 9): A first linear regression model including the experimental condition (-1 = Vege Whopper; 1 = New Whopper) finding the burger appetizing; the probability of actually ordering this meal; gender (-1 = Female; 0 = Other; 1 = Male) age; the number of hours since the last meal; the frequency of visits to burger king (1 = I never go there; 2 = Less than once a year; 3 = Once a year; 4 = Once every six months; 5 = Once every quarter; 6 = Once a month; 7 = Once a week 8 = Several times a week) and diet plan (1 = vegan; 2 = vegetarian; 3 = flexitarian; 4 = omnivorous) were achieved with the caloric estimation score as a dependent variable (general model: F(8,894) = 7.53, p < .001, R 2 adj = 0.06). The analysis revealed a significant effect of diet plan, b = 0.13, p < .001; the probability of 10): Which adds to the first the interaction of the experimental condition with diet plan (general model F(9,893) = 6.82, p < .001, R 2 adj = 0.06). The analysis revealed a significant effect of the diet plan, b = 0.14, p < .001; the probability of actually ordering this meal, b = 0.09, p = .011 and age, b = -0.12, p < .001. The analysis also shows a tendential effect of the frequency of visits to Burger King, b = 0.06, p = .079. However, no effects were found for the experimental condition, b = -0.19, p = .332; for finding the burger appetizing,


Lionel Birglen
email: lionel.birglen@polymtl.ca

Marc Gouttefarde
email: marc.gouttefarde@lirmm.fr

Stiffness of Planar 2-DOF 3-Differential Cable-Driven Parallel Robots

Keywords: Cable-driven parallel robots, differential pulley actuation, stiffness analysis

Planar 2-degree-of-freedom (DOF) 3-differential Cable-Driven Parallel Robots (CDPRs) consist of a point-mass end-effector driven by a number of cables. Each cable is divided into four segments, three of them being connected to the point-mass end-effector by means of routing pulleys. This paper deals with the stiffness analysis of such planar 2-DOF 3-differential CDPRs. Based on the usual linear spring cable elongation model, the expression of the stiffness matrix is derived. The stiffness and workspace of several examples of planar 2-DOF 3-differential CDPRs are then compared. The results of these comparisons illustrate that the stiffness of planar CDPRs can be significantly improved by means of pulley differentials.

Introduction

This paper deals with cable-driven parallel robots (CDPRs) whose cable segments are not each driven by a single actuator but are coupled through differentials. Coupling and transmitting the actuation torque of a single actuator to several cable segments has been shown in previous works to be a possible solution to extend the workspace of CDPRs by purely passive mechanical means [START_REF] Khakpour | Synthesis of differentially driven planar cable parallel manipulators[END_REF][START_REF] Khakpour | Workspace augmentation of spatial 3-dof cable parallel robots using differential actuation[END_REF][START_REF] Khakpour | Analysis and optimization of a new differentially driven cable parallel robot[END_REF], i.e. without relying on additional actuators or relocating the latter on a structural frame. Compared to conventional CDPRs, more cable segments then cluttered the workspace but this should not be a critical issue for planar and suspended CDPRs. In the specific example studied here, a single actuator simultaneously controls the lengths of three cable segments going from the ground to the point-mass end-effector of a planar robot and the coupling between these cable segments is implemented through pulley differentials, similarly in principle to the well-known block and tackle. The present paper presents and quantifies the stiffness of such planar 2-DOF 3-differential CDPRs. Such differential couplings will be shown to provide a noticeable improvement in the stiffness of the CDPR.

As shown in Fig. 1, each cable of the CDPR consists of four successive segments B i P , P E i , E i M i , and M i P . Three of these cable segments link the point mass P , which plays the role of the CDPR 2-DOF end-effector, to the ground. In the remainder of this paper, these three segments are referred to as the active cable segments. The other one, E i M i , extends between points E i and M i which are both fixed to the ground. In other words, cable i exits the base first at point B i , and then goes in this order through points P , E i , M i to finally be attached to point P . Hence, these pulley drives are referred to as 3-differential since each one has three "active" outputs. Figure 2 shows this cable routing. In order to implement such a mechanism, pulleys need to be used at points P , E i , and M i . However, as a first approximation, the diameters of these pulleys are neglected (considered to be null) and pulley friction is ignored (no friction). The latter assumption implies that the cable tension is constant all along the cable, i.e., the four cable segments have the same tension and each active segment applies a force of identical magnitude on the end-effector. Let n be the number of cables used to drive point P (n = 3 in the case of Fig. 1). This paper deals with the case of planar 2-DOF 3-differential CDPRs with n ≥ 2 cables. The mass of the cables is neglected and all the cable segments are considered to be straight line segments.

Stiffness and vibration analyses of usual CDPRs, where the driving cables are directly attached to the end-effector, have been proposed in a number of works, e.g. [START_REF] Behzadipour | Stiffness of cable-based parallel manipulators with application to stability analysis[END_REF][START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF][START_REF] Arsenault | Workspace and stiffness analysis of a three-degree-of-freedom spatial cable-suspended parallel mechanism while considering cable mass[END_REF][START_REF] Yuan | Static and dynamic stiffness analyses of cable-driven parallel robots with non-negligible cable mass and elasticity[END_REF][START_REF] Yuan | Vibration analysis of cable-driven parallel robots based on the dynamic stiffness matrix method[END_REF][START_REF] Anson | Orientation workspace and stiffness optimization of cable-driven parallel manipulators with base mobility[END_REF][START_REF] Abdolshah | Optimizing stiffness and dexterity of planar adaptive cable-driven parallel robots[END_REF][START_REF] Jamshidifar | Kinematicallyconstrained redundant cable-driven parallel robots: Modeling, redundancy analysis, and stiffness optimization[END_REF]. As previously mentioned, the main difference with these previous works is that the present paper deals with the stiffness analysis of planar point-Fig. 2. Cable routing in a single differential: The cable starts at point Bi where the actuated drum is located, goes to P (the end-effector), then to Ei where it is redirected to point Mi in order to be finally attached to point P .

mass CDPRs with 3-differential pulley drive. To the best of our knowledge, the only previous work on stiffness analysis of differential-pulley planar CDPRs is [START_REF] Nelson | On improving stiffness of cable robots[END_REF] where the case of cables with two active segments is considered. Moreover, the stiffness analysis in [START_REF] Nelson | On improving stiffness of cable robots[END_REF] only considered stiffness in a single direction and did not mention the trade-off between the size of the workspace and the improvement in stiffness inherent to pulley differentials as will be shown here.

This paper is organized as follows: First, Section 2 presents the Jacobian and wrench matrices of differential-pulley-driven planar point-mass CDPRs with three active cable segments. Then, in Section 3, the stiffness matrix of these mechanisms is derived based on the usual linear spring cable elongation model. Finally, Section 4 reports simulations of the resulting stiffness of several examples of planar 2-DOF 3-differential CDPRs.

Wrench and Jacobian Matrices

The active length of the cable i in each differential routing is defined as l i = l BiP +l EiP +l MiP where l BiP = --→ B i P , l EiP = --→ E i P and l MiP = --→ M i P are the strained lengths of the active cable segments B i P , E i P and M i P , respectively. Length l i is the sum of the cable segment lengths that change when the position of point P changes. It should not be confused with the total length of the cable which includes the length of the cable segment between points E i and M i . Let us define the vector l of the active cable lengths l i as l = [l 1 , l 2 , . . . , l n ]

T . Moreover, let ˙l be the time derivative of l, p = [p x , p y ] T be the position vector of point P in the fixed reference frame (0, x, y), and ṗ the velocity of point P . A Jacobian matrix J then maps ṗ (resp. dp) into ˙l (resp. dl)

˙l =      l1 l2 . . . ln      = J ṗ ⇐⇒ dl = Jdp (1)
with

J =    u T B1 + u T E1 + u T M 1 . . . u T Bn + u T En + u T M n    n×2 (2)
where u Bi , u Ei and u M i are the unit vectors directed along the cable segments B i P , E i P , and M i P , pointing from the base points B i , E i , and M i , to point P , respectively. The vector of cable tensions t i is denoted t = [t 1 , t 2 , . . . , t n ] T . Pulley friction being neglected, the cable tension t i is constant all along the cable. The force f applied by the n cables to the point-mass P is then given by

f = Wt (3)
where the 2 × n wrench matrix W is defined as W = -J T .

3

Stiffness Analysis

The elastic potential energy of cable i is given by

V i = 1 2 k i (l ti -l 0i ) 2 (4)
where l ti is the total length of cable i

l ti = l i + l EiMi (5)
with l EiMi the length of the passive cable segment E i M i and l i = l BiP + l EiP + l MiP the active length of cable i, as defined in Section 2. l 0i is the corresponding unstrained length, l 0i = l 0 B i P +l 0 E i P +l 0 E i M i +l 0 M i P . The elongation is supposed to be uniform along the cable length and the cable is assumed to be a linear spring with

t i = k i (l ti -l 0i ), k i = AE l0i
, where A and E are the cable cross-sectional area and elastic modulus, respectively.

The total elastic potential energy V of the 3-differential CDPR is then the sum of the elastic potential energies of all its n cables

V = n i=1 V i = 1 2 n i=1 k i (l ti -l 0i ) 2 (6)
Computing the derivative of Equation ( 6) yields

dV = n i=1 dV i = n i=1 k i (l ti -l 0i ) dl ti = n i=1 k i (l ti -l 0i ) dl i (7)
where dl ti = dl i according to Eq. ( 5) and to the fact that dl EiMi = 0 (the distance between the ground points E i and M i is constant). Then, defining the diagonal matrix D and vectors l t and l 0 as follows

D =    k 1 . . . k n    , l t =      l t1 l t2 . . . l tn      and l 0 =      l 01 l 02 . . . l 0n      (8)
Eq. ( 7) can be written

dV = (l t -l 0 ) T Ddl (9)
According to Eq. ( 1), dl is equal to Jdp so that according to Eq. ( 9)

dV dp =     ∂V ∂p x ∂V ∂p y     = J T D (l t -l 0 ) (10)
where it can be noted that D (l t -l 0 ) = t since t i = k i (l ti -l 0i ). The stiffness matrix K of the 3-differential CDPR is defined as the second derivative of the elastic potential energy, namely

K = d 2 V dp 2 =      ∂ 2 V ∂p 2 x ∂ 2 V ∂p x ∂p y ∂ 2 V ∂p y ∂p x ∂ 2 V ∂p 2 y      (11)
Taking the derivative of Eq. ( 10) with respect to p, the following classic expression of the stiffness matrix is obtained

K = J T D dl dp + dJ T dp D (l t -l 0 ) = J T DJ + dJ T dp t (12)
where, once again, the equality dl t = dl has been used. It is worth noting that this stiffness matrix K can only be used in local stiffness analyses because Eq. ( 12) is obtained from Eq. ( 10) by assuming that l 0 is constant. In other words, the matrix K is defined at an equilibrium, defined by both the position p and the cable tensions t, and it can be used to analyze the stiffness of the robot in the vicinity of this equilibrium. In this local analysis, the actuators are assumed to be locked or their position controlled to a constant value. Note also that, according to Eq. ( 3) and [START_REF] Abdolshah | Optimizing stiffness and dexterity of planar adaptive cable-driven parallel robots[END_REF], dV dp = J T t = -f , and thus

K = d dp dV dp = d(-f ) dp ⇐⇒ Kdp = d(-f ) (13)
Hence, in the vicinity of an equilibrium defined by p and t, K maps an infinitesimal displacement dp in the position of point P to the corresponding infinitesimal change in the force -f , where f is the force applied by the cables to point P (Eq. ( 3)). The approach used above to derive the stiffness matrix is a rather classic one, but care must be taken to use in Eq. ( 12) the Jacobian matrix defined in Eq. ( 1), where each row of this matrix is equal to the sum u T Bi + u T Ei + u T M i of the unit vectors directed along the three straight line segments delineated by the active cable segments.

Referring to the right-hand side of Eq. ( 12), the stiffness matrix is seen to be the sum of two matrices. In the next section of this paper, for simplicity and due to space limitations, only the influence on stiffness of the matrix J T DJ will be considered. This section reports simulations of the stiffness of six planar 2-DOF 3differential CDPRs with n = 3 cables, across their respective workspaces. These six examples are defined as Cases 1 to 6 in Tables 1 and2. Case 1 is shown in Fig. 3 together with its Wrench-Closure Workspace (WCW) [START_REF] Gouttefarde | Analysis of the wrench-closure workspace of planar parallel cable-driven mechanisms[END_REF] and Wrench-Feasible Workspace (WFW) [START_REF] Bosscher | Wrench-feasible workspace generation for cable-driven robots[END_REF]. Here, the WFW is defined as the set of positions of point P such that any force f of magnitude less or equal to 0.1 N can be generated by the cables at P with tensions t i , i = 1 . . . n, satisfying t min ≤ t i ≤ t max where t min = 0.1 N and t max = 1 N.

Results

Case 2 defined in Table 1 is a planar 2-DOF 3-differential CDPR where the three active cable segments are superposed. It is obtained from Case 1 by taking B i ≡ E i ≡ M i for i = 1, 2 and 3. Case 3 in Table 1 defines a CDPR without pulley differential where each cable consists of only one segment from B i to P . It is obtained from Case 1 by keeping segment B i P and removing the other cable segments P E i , E i M i , and M i P .

Case 4 defined in Table 2 is another planar 2-DOF 3-differential CDPR. It is shown in Fig. 4 where its WFW is defined as above for Case 1. Case 5 is a planar 2-DOF 3-differential CDPR obtained from Case 4 by superposing the 3

Case 1 Case 2 Case 3 x (m) y (m) x (m) y (m)
x (m) y (m) B1 cos(π/3) sin(π/3) cos(π/3) sin(π/3) cos(π/3) sin(π/3) E1 cos(2π/3) sin(2π/3) cos(π/3) sin(π/3) active segments of each cable and Case 6 is a CDPR without pulley differential obtained from Case 4 by removing, for each cable, all segments but B i P .

× × M1 0 1 cos(π/3) sin(π/3) × × B2 -1 0 -1 0 -1 0 E2 cos(-2π/3) sin(-2π/3) -1 0 × × M2 cos(-5π/6) sin(-5π/6) -1 0 × × B3 cos(-π/3) sin(-π/3) cos(-π/3) sin(-π/3) cos(-π/3) sin(-π/3) E3 1 0 cos(-π/3) sin(-π/3) × × M3 cos(-π/6) sin(-π/6) cos(-π/3) sin(-π/3) × ×
y (m) x (m) y (m) x (m) y (m) B1 1 0 1 0 1 0 E1 cos(2π/3) sin(2π/3) 1 0 × × M1 cos(π/3) sin(π/3) 1 0 × × B2 cos(2π/3) sin(2π/3) cos(2π/3) sin(2π/3) cos(2π/3) sin(2π/3) E2 cos(-2π/3) sin(-2π/3) cos(2π/3) sin(2π/3) × × M2 -1 0 cos(2π/3) sin(2π/3) × × B3 cos(-2π/3) sin(-2π/3) cos(-2π/3) sin(-2π/3) cos(-2π/3) sin(-2π/3) E3 1 0 cos(-2π/3) sin(-2π/3) × × M3 cos(-π/3) sin(-π/3) cos(-2π/3) sin(-2π/3) × ×
As mentioned at the end of Section 3, the subsequent stiffness analysis is based on considering that the stiffness matrix is K = J T DJ. To "normalize" the stiffness coefficient with respect to specific values, we use E = 1 Pa and A = 0.001 m 2 . Note that these values, together with the t min and t max values defined above, do not correspond to actual physical values since, for instance, steel wire ropes typically have elastic modulus E of tens of GPa. However, it is not important for the purposes of this section where stiffness comparisons between several 2-DOF 3-differential CDPRs are made. Moreover, in the calculation of D, for simplicity, we assume that the unstrained length of each cable segment is equal to its strained length. For a given position of the end-effector P , the two eigenvalues of the 2 × 2 stiffness matrix K can then be calculated. The minimum (resp. maximum) stiffness values shown in the left part (resp. right part) of Fig. 5 are defined as the minimum (resp. maximum) of these two eigenvalues. The values of the ratios of the minimum to the maximum eigenvalues over the WCW of Case 1 are shown in Fig. 6.

The means of the minimum stiffness values, maximum stiffness values and ratio of minimum to maximum stiffness values over the WCW of the planar 2-DOF 3-differential CDPR examples defined in Tables 1 and2 are given in Table 3. The mean of the minimal stiffness in Case 1 is 2.4 times larger than in Case 3, where Case 3 corresponds to a classic CDPR with no pulley differential. The means of the maximum stiffness and stiffness ratio in Case 1 are both 1.4 times larger than in Case 3. Comparing Case 2 to Case 3, the minimum and maximum stiffness values are 3 times larger for Case 2 while the stiffness ratios are identical. Hence, planar 2-DOF 3-differential CDPRs (Cases 1 and 2) present a larger stiffness over their workspace than the corresponding CDPR without Finally, Table 4 shows a comparison of the size of the WCW and WFW in each case. The WFW is defined for all the six cases as it has been defined above for Case 1. From these examples, it can be concluded that the cases of 2-DOF 3-differential CDPRs with superposed cable segments (Cases 2 and 5) lead to the largest stiffness improvements but with a marginal gain on the sizes of the WCW and WFW, as compared to the cases without pulley differentials (Cases 3 and 6). In comparison, the stiffness improvement is less significant in the cases of 2-DOF 3-differential CDPRs with distinct cable segments (Cases 1 and 4) with, however, a slightly better improvement in the WCW and WFW sizes. This total area is the one delimited by the circles shown in Fig. 3 and4.

also showed that a trade-off may exist between designing a CDPR with pulley differentials to improve stiffness and designing one to increase the workspace size.

Fig. 1 .

 1 Fig. 1. A planar 2-DOF 3-Differential CDPR where each cable consists of four segments BiPi, PiEi, EiMi, and MiPi, three of these segments connecting the end-effector P to the ground points Bi, Mi and Ei.

Fig. 3 .

 3 Fig. 3. Example planar 2-DOF 3-differential CDPR with 3 cables (Case 1)

Fig. 4 .

 4 Fig. 4. Example planar 2-DOF 3-differential CDPR with 3 cables (Case 4)

Fig. 5 .

 5 Fig. 5. Minimum and maximum stiffness values over the WCW of Case 1

Fig. 6 .

 6 Fig. 6. Ratio of minimal to the maximal stiffness values over the WCW of Case 1

Table 1 .

 1 Geometries of the first three examples of planar 2-DOF 3-differential CDPRs. The crosses × indicate that the corresponding point Ei or Mi does not exist, i.e., Case 3 defines a CDPR without pulley differential where each cable consists of only one segment from Bi to P .

	Case 4	Case 5	Case 6
	x (m)

Table 2 .

 2 Geometries of the last three examples of planar 2-DOF 3-differential CDPRs. Case 6 defines a CDPR without pulley differential where each cable consists of only one segment from Bi to P .

Table 3 .

 3 Means of the minimum stiffness values (min), maximum stiffness values (max) and ratio of minimum to maximum stiffness values (ratio) over the WCW of the 2-DOF 3-differential CDPR examples pulley differential (Case 3). Similar results are obtained by comparing Cases 4 and 5 to Case 6. The cases with superposed cable segments (Cases 2 and 5) have larger minimum and maximum stiffness values than the ones with distinct cable segments (Cases 1 and 4). Note that Case 4 has the best stiffness ratio.

	1	0.0018	0.0041	0.43
	2	0.0022	0.0086	0.31
	3	0.00074	0.0029	0.31
	4	0.0012	0.0016	0.79
	5	0.0022	0.0086	0.31
	6	0.00074	0.0029	0.31

Table 4 .

 4 WCW and WFW coverage of the total area occupied by the CDPR examples.

	Case WCW WFW
	1 37.6% 29.5%
	2 32.4% 27.3%
	3 32.4% 24.7%
	4 33.6% 22.6%
	5 32.4% 27.3%
	6 32.4% 24.7%