Design of Controller for Crawling to Sitting Behavior of Infants

Transcription

1 Design of Controller for Crawling to Sitting Behavior of Infants A Report submitte for the Semester Project To be accepte on: 29 June 2007 by Neha Priyaarshini Garg Supervisors: Luovic Righetti Prof. Auke J. Ijspeert Biologically Inspire Group of Robotics Department of Computer Science Swiss Feeral Institute of Technology Lausanne, Switzerlan

2 Acknowlegements First an foremost, I woul like to thanks Luovic Righetti for his help throughout this project, particularly in constructing equations of Dynamical system an feeback on the report. I am also thankful to Professor Auke Jan Ijspeert for allowing me to o my semester project with BIRG, motivating me by showing interest in the project an giving useful suggestions for the project. 2 7/2/2007

3 TABLE OF CONTENTS 1 INTRODUCTION PROBLEM DESCRIPTION AND MAIN OBJECTIVES PROPOSED SOLUTION THE ROBOT ORGANIZATION OF THE REPORT DEVELOPMENT OF HAND MADE TRAJECTORY OBSERVATIONS FROM REAL INFANTS THE HAND-MADE TRAJECTORY ANALYSIS OF MAIN CHARACTERISTICS OF TRAJECTORY DIVISION INTO TWO PHASES ROBUSTNESS OF SITTING TRANSITION POSITION OF CENTER OF MASS AS STABILITY CRITERION SIGNIFICANCE OF SPEED OF TORSO CONCLUSION OF ANALYSIS DYNAMICAL SYSTEM FOR SITTING NEED OF DYNAMICAL SYSTEM AND MAIN CHALLENGES MATHEMATICAL SYSTEM FOR SITTING TRAJECTORY SWITCHING FROM CRAWLING TO SITTING CONCLUSION AND FUTURE WORK REFERENCES /2/2007

4 Chapter 1 Introuction 1.1 Problem Description an Main Objectives The RobotCUB European project [1] aims to stuy cognitive abilities of a chil by builing a 2 year ol infant-like humanoi robot name icub. The project has two main goals: first, to create an open an freely available humanoi platform for research in emboie cognition, an secon, to stuy cognitive evelopment. To achieve the goal of cognitive evelopment, it is essential that the robot is able to explore its environment by crawling an sitting just like infants. Therefore, as a part of RobotCUB project, BIRG, EPFL is esigning a controller for the locomotion of the robot so that it is able to crawl on its hans an knees just like an infant an shoul be able to make transition from crawling to sitting position an vise versa. Controller for locomotion of robot in unpreictable environments is quite a challenging task. The main strength of locomotion of infants lies in the fact that infants are able to o locomotion robustly in almost any kin of environment. It may be possible to esign very efficient controllers for locomotion when the external environment is known but such controllers o not have the capability of robust locomotion in new an unpreictable environment. Therefore, unerstaning the biological mechanisms of locomotion is the key for evelopment of controllers for robust locomotion of autonomous robots. The unerlying biological mechanisms by which infants are able to crawl an sit in almost any kin of environment can be learne by moeling the behavior of infants. A controller for crawling behavior of robot has alreay been evelope in [2] by observation of crawling in real infants. It is base on CPGs which are neural circuits responsible for locomotion foun in spinal cor of animals. To esign the controller first the crawling behavior of infants was stuie an important characteristics of the trajectory were extracte. Then a mathematical moel of CPGs base on couple nonlinear oscillators was evelope to reprouce the crawling gait of infants. The goal of this project is to stuy the transition from crawling to sitting in infants an to finally esign a controller for the robot that woul enable it to perform this transition in the same way as infants. All the experiments on the robot are one in Webots, a simulation software [3]. 1.2 Propose Solution For esigning a controller for crawling to sitting, a biologically inspire approach is followe. - First, the transition from crawling to sitting in real infants is stuie qualitatively. - From the qualitative observation of real infants sitting, a han-mae trajectory for the DOFs of the robot is esigne such that the robot can sit. 4 7/2/2007

5 - Then the main characteristics of the trajectory are stuie. For example: Is the position of the projection of the center of mass of the robot on the groun, always in the support polygon? When is the robot unstable? What is the state of the robot when entering these instability regions? What are the critical times of the movement? - After this, the trajectories are optimize to increase the stability of the transition - After an optimum trajectory is mae, a ynamical system can be implemente for the transition from crawling to sitting by integrating it with the previously esigne controller for crawling [2]. 1.3 The Robot The etaile specifications of the real ICUB robot have been given in [4]. The Webots moel of ICUB has 27 egrees of freeom: 6 in the hea (3 in the neck for tilt, swing an pan an 3 in the eyes), 5 in each leg (3 in hip joint for flexion/extension, abuction/auction an rotation an one each in knee an ankle for flexion/extension), 4 in each arm (3 in shouler for flexion/extension, abuction/auction an rotation an one in elbow for flexion/extension) an 3 in the torso (roll, pitch an yaw) (Fig 1.1). The real ICUB has 53 egrees of freeom with extra egrees of freeom in han an wrist but for simulation of crawling an sitting behaviors the 27 egrees of freeom in the Webots moel of ICUB are sufficient. Shouler Joint: 3 DOF Torso Joint: 3 DOF Hip Joint: 3 DOF Elbow Joint: 1 DOF (Flexion / Extension) Knee Joint: 1 DOF (Flexion / Extension) Y Z Ankle Joint: 1 DOF (Flexion / Extension) X Figure 1.1: The egrees of freeom in the Webots moel of the ICUB 5 7/2/2007

6 The range of motion of joint angles (in raians) use for sitting an crawling along with the names use for those angles in the report is shown in Figure1.2. DOF: Shouler (Flex/Ext.) Angle name: Right_arm_1 /Left_arm_1 Angle range: to 4.01 DOF: Torso Roll Angle name: Torso_1 Angle range: to 1.57 DOF: Hip (Flex/Ext) Angle name: Right_leg_1 / Left_leg_1 Angle range: to 1.75 DOF: Elbow (Flex/Ext) Angle name: Right_elbow / Left_elbow Angle range: 0.0 to _ + _ DOF: Ankle (Flex/Ext) Angle name: Right_ankle / Left_ankle Angle range: to DOF: Knee (Flex/Ext) Angle name: Right_knee / Left_knee Angle range: to Back View Left Right DOF: Shouler Abuc/Auc Angle name: Left_arm_2 Angle range: to DOF: Shouler Rotation Angle name: Left_arm_3 Angle range: to 1.57 DOF: HipAbuc/Auc Angle name: Left_leg_2 Angle range: to DOF: Hip Rotation Angle name: Left_leg_3 Angle range: to _ + + DOF: Torso Pitch Angle name: Torso_2 Angle range: to 1.22 DOF: Shouler Abuc/Auc Angle name: Right_arm_2 Angle range: to 2.62 DOF: Shouler Rotation Angle name: Right_arm_3 Angle range: to 1.57 DOF: HipAbuc/Auc Angle name: Right_leg_2 Angle range: to 0.82 DOF: Hip Rotation Angle name: Right_leg_3 Angle range: -1.4 to Figure 1.2: Names an range of motion of joint angles of Webots moel of the ICUB 6 7/2/2007

7 1.4 Organization of the report In chapter 2, I will escribe the qualitative characteristics of the transition from crawling to sitting observe in real infants an compare the han-mae trajectory evelope for the robot with the trajectory followe by real infants. Then, I will analyze the main characteristics of the han-mae trajectory an explore the various stability criteria for the trajectory. In chapter 3, I will escribe how we can switch from crawling to sitting controller when an external signal is provie an how we can generate the trajectories for sitting from a ynamical system. Chapter 2 Development of han-mae trajectory 2.1 Observations from real infants The stuy of crawling to sitting in real infants was one on the kinematical ata of crawling babies provie by Uppsala University (Sween) to BIRG, EPFL. The ata contains recore vieos of about 1 year ol infants crawling on the groun, shifting to sitting from crawling an from crawling to sitting. From the vieos, it was observe that infants generally sit in two ways as shown in Figure 2.1. In the first way (Figure 2.1a), infants sit on their legs while in the secon way (Figure 2.1 b), infants sit on their hips. The first metho of sitting is not aequate for robots as it puts lot of pressure on knee an ankle joints. Also the knee nees to be flexe towars hip to a large extent which is not achievable in the real icub robot. Therefore, the secon metho of sitting has been stuie for the evelopment of han-mae trajectory. (a) (b) Figure 2.1: The two ways in which infants generally sit When infants sit on their hips, two main characteristics of the transition can be notice: 7 7/2/2007

8 (i) (ii) First, infants bring one of their leg forwar using the other leg an arms as a support (First 3 snapshots of real infants in Figure 2.2) Then they move their arm back an finally sit on their hip(last 3 snapshots of real infants in Figure2.2) Figure 2.2: Infants oing transition from crawling to sitting The main focus while eveloping han-mae trajectory has been on the capturing of these two main characteristics of the transition from crawling to sitting in infants. In Section 2.2, I will escribe the han-mae trajectory evelope for the ICUB. 2.2 The han-mae trajectory The trajectory is evelope by specifying the angles for Relevant DOFs after 5 TimeSteps (1 Time Step = 64 ms) an then interpolating between these angles for every TimeStep using PCHIP (Piecewise Cubic Hermite Interpolating Polynomial) [5]. This interpolation metho gives an interpolating polynomial whose first orer erivative is continuous. Unlike spline interpolation, the secon orer erivative is not continuous but the shape of ata is preserve by this interpolation i.e the interval where the ata is monotonic, interpolating polynomial is also monotonic (Figure 2.3). The sitting metho of the robot using the han-mae trajectory has been compare with the sitting metho of real infant in Figure 2.4. The two main characteristics of the sitting metho that were observe in real infants have been capture in the han mae trajectory also. To bring one of the legs forwar (left leg in this case) first the DOF Torso Pitch is move which shifts the weight of the boy to the right sie (Fig 2.5 :First snapshot) Then the DOF right leg (Abuction / Auction an Rotation) move which broaens support polygon i.e right half of the boy starts supporting the boy (Fig 2.5 :Secon snapshot). Also simultaneously left knee starts extening so that it can come forwar. While the left 8 7/2/2007

9 knee is extening, DOF left leg (Abuction/Auction an Rotation) move. This movement of left leg helps in extension of left knee without putting much pressure on the right half of the boy. If the DOF left leg (Abuction / Auction an Rotation) o not move, the extension of knee will increase the height of left half of the boy an robot may fall towars the right sie. After the extension of left knee is one the DOF left leg (Abuction / Auction an Rotation) move back to their original state an the first phase of bringing left leg forwar is complete (Fig 2.5 :thir snapshot). Then the right arm moves so that the robot is able to sit on the hip (Fig 2.5 : Last 3 snapshots). Figure 2.3: Comparison of interpolation of ata using Pchip an spline interpolation Figure 2.4: Han mae trajectory of the robot compare with real infant 9 7/2/2007

10 Figure 2.5: Main actions of robot while sitting. The re lines shown the support polygon an the green box shows the projection of center of mass on the groun. Though the main characteristics of sitting transition in infants have been capture, there are some visible ifferences in the sitting metho of infant an the robot because of the constraints on the maximum angle up to which hip joint of the robot can be extene. Real infant exten their legs to a large extent while moving one of their leg forwar (Angles right_leg_1 an left_leg_1 as efine in Figure 1.2 are at least 2 raians). Because of this, the hips of the infant are almost touching the groun when the infant has brought its leg forwar (First 3 snapshots of real infant in Figure 2.4) but the robot cannot exten its legs to such a large extent. The maximum range of angles right_leg_1 an left_leg_1 is 1.75 raians. Due to this constraint, hips of the robot are quite above from the groun level when it starts moving its arm back (last 3 snapshots of the robot in Figure 2.4) as compare to the real infant. This makes the robot more unstable in the secon part of the transition. In the next section, I will analyze the main characteristics of the han-mae trajectory like its stability, robustness, position of projection of center of mass etc. 2.3 Analysis of main characteristics of trajectory Division into two phases As observe in the real infant, the transition from crawling to sitting can be ivie into two parts. In the first part, chil brings one of its legs to the front by using secon an thir egrees of freeom of leg. After that, chil moves its arm backwar in orer to sit on the groun. The variations of joint angles of relevant DOFs of robot while sitting are shown in Figure2.6. The yellow line (left most) in the plots correspons to the point when robot starts rotation of its leg in orer to bring it forwar (left leg in this case). The orange line (mile) correspons to the point when robot has complete the rotation of its left leg an starts moving its right arm so that it can sit. The re line (right most) correspons to the point when robot sits on the groun. If the trajectory is specifie only until orange line (mile), then robot oes not sit but is stable at its position an right arm can be move at any time to make the robot sit. Therefore there is a clear istinction between two phases. The projection on the groun of the center of mass of robot is insie the support polygon uring the first phase an goes outsie the support polygon uring the secon phase. We can say that the secon phase is the critical perio while making transition from crawling to sitting because uring this perio the robot is unstable. This is also inicate by the 10 7/2/2007

11 torso spee of the robot (plotte in soli black line with the variation of joint angles with time in Figure 2.6) which goes to its maximum value uring the secon phase. In the next section, I will explain the metho of checking the robustness of the sitting transition using the critical perio to limit the parameter space an analyze the robustness of the current trajectory Robustness of sitting transition The transition from crawling to sitting has to be robust to perturbation as robots eviate from the specifie trajectories very often. The robustness of the trajectory cannot be checke by just varying the points specifie for the trajectory of a egree of freeom of the robot one at a time an seeing whether robot falls or not because trajectory points are interepenent. The robot may or may not fall for a given value of a trajectory point of a egree of freeom epening upon the values specifie for the other trajectory points. Varying a trajectory point for all possible values of other trajectory points is not possible because the number of possible combinations grow exponentially with the number of trajectory points. Therefore we nee to limit the trajectory points that we nee to vary. The notion of critical perio in the sitting transition efine in section can be use for this purpose. The robot is very stable in the first phase of the transition as one of its legs an both of its arms are touching the groun an center of mass is well insie the support polygon. Therefore we can check robustness of the trajectory only in the critical perio when the robot is unstable. The egrees of freeom which move uring the critical perio are: Right arm (Flexion / Extension an Abuction / Auction ) Torso (Roll an Pitch) To check the robustness of the sitting transition, I varie the trajectories of these DOFs (one at a time) in the critical perio i.e. I varie the two specifie trajectory points(point no. 6 an 7, Figure 2.7(left)) that fall between orange an re line. Angle (raians)/ Spee (m/s) Angle (raians)/ Spee (m/s) Time (in sec) Time (in sec) 11 7/2/2007

12 Angle (raians)/ Spee (m/s) Angle (raians)/ Spee (m/s) Time (in sec) Time (in sec) Angle (raians)/ Spee (m/s) Time (in sec) Figure 2.6: Variation of joint angles of relevant egrees of freeom an torso spee (vertical in soli black line an absolute in otte black line) with time. (Top left) Variation of angles Right_arm_1 an Right_arm_2. (Top Right) Variation of angles Left_arm_1 an Left_arm_2. (Mile Left) Variation of angles Right_leg_1, Right_leg_2, Right_leg_3 an Right_knee. (Mile Right) Variation of angles Left_leg_1, Left_leg_2, Left_leg_3 an Left_knee. (Bottom) Variation of angles Torso_1 an Torso_2. The yellow vertical line (leftmost) is rawn at the time when robot starts moving its left leg. The orange vertical line (mile) is rawn at the time when robot has brought its left leg forwar an starts moving its right arm. The re line (rightmost) is rawn at the time when robot sits on the groun. The names of the angles whose variations are plotte are efine in Figure 1.2. The legs remain almost static uring the critical perio. Therefore, for right leg (Abuction / Auction an Rotation) the constant value which these DOFs have uring 12 7/2/2007

13 critical perio is varie by varying Point no. 3 (Fig. 2.7 (right)) in the trajectory of right leg (Abuction/ auction, angle name right_leg_2) an Point no. 4 (Fig. 2.7 (right)) in the trajectory of right leg (Rotation, angle name right_leg_3). Other DOFs like right leg (Flexion /Extension), right knee, left leg (Flexion / Extension) an left knee have not been varie because by varying these DOFs, robot is able to sit but uring transition there is lot of pressure on the ankles as knees o not touch the groun in most of the cases. The left leg (Abuction / Auction an Rotation) have also not been varie because their value has to be near zero i.e. left leg has to come forwar for the first phase to en, only then we can enter critical perio by moving right arm backwars. After the selection of the trajectory points that have to be varie, next important thing that nees to be etermine is the etection of robot fall. The robot is consiere to have fallen when: (i) (ii) the hea of the robot touches the groun. The angle of the line joining torso an hea with the vertical axis is greater than 45 after the hips have touche the groun. The secon criterion is useful because sometimes robot oes not fall with hea touching the groun but is sitting in very unstable position from where a small perturbation may make the robot fall. Figure show the range of angles of the trajectory points over which robot oes not fall for relevant DOFs. From Figure , we can observe that there is a well efine an quite large region for all the DOFs in which robot is able to sit with stability. The specifie points in the trajectory shoul lie in the center of that region so that sitting proceure is least affecte by perturbations. This is true for the specifie trajectory points of the han mae trajectory. In the next section, I will analyze the use of projection of center of mass as a criterion for eciing the stability of sitting transition. Point 6 Point 3 Point 7 Point 4 Time (in sec) Figure 2.7: The points of trajectory that are varie Time (in sec) 13 7/2/2007

14 Figure 2.8: Falling behavior of robot on varying the points of trajectory specifie for right arm in the critical region (left: right arm Flexion / Extension (angle name: right_arm_1), right: right arm Abuc / Auc (angle name right_arm_2)) ( Falls = 1 when robot falls) Figure 2.9: Falling behavior of robot on varying the points of trajectory specifie for torso in the critical region ( left : torso Roll (Angle name torso_1), right: torso Pitch (Angle name torso_2)) (( Falls > 0 when robot falls) For torso_2, when the robot falls on the hea, falls has been given value 2 an when the sitting is not stable, falls has been given value 1. Figure 2.10: Falling behavior of robot on varying the constant value of trajectory specifie for right leg in the critical region (Left: right leg Abuction / Auction (angle name: right_leg_2), Right: Right leg Rotation (angle name: right_leg_3)) ( Fall = 1 when robot falls) 14 7/2/2007

15 2.3.3 Stability criterion base on center of mass To classify which sitting transitions are goo an which are ba, a criterion is neee to measure the stability of the sitting transition. The position projection of center of mass on the groun provies goo information about the stability of the robot. If the projection of center of mass on the groun is outsie the support polygon, the robot is unstable at that moment otherwise we can consier it stable when the spee of center of mass is not very high which is generally the case in sitting transition. Also if the istance of projection of center of mass from support polygon is larger, the instability is higher. Thus both the time for which center of mass is outsie support polygon an the istance of center of mass from the support polygon, inicate the stability of the robot. To capture both these quantities, I efine stability measure as integration of istance of center of mass from support polygon with time uring sitting. The istance of center of mass from support polygon is taken as perpenicular istance of center of mass from the closest ege of support polygon when center of mass is outsie support polygon an zero when the center of mass is insie support polygon. Figure show the variation in this stability measure when the trajectory points of the egrees of freeom are varie like in Section The stability measure has been name CM Distance in the plots. As can be seen from the plots, the CM Distance cannot be use to istinguish between the regions of robot falling an not falling. This means that for sitting the center of mass always goes outsie the support polygon. Also, there is not much variation in this quantity for the region in which robot oes not fall. Therefore this criterion cannot be use to classify sitting transitions as goo or ba. Figure 2.11: The left plot shows the region where robot falls (/ oes not fall ) an the right plot shows the variation in CM_Distance on varying the Points 6 an 7 of the trajectory of Right Arm Flexion / Extension. 15 7/2/2007

16 Figure 2.12: The left plot shows the region where robot falls (/ oes not fall) an the right plot shows the variation in CM_Distance on varying the Points 6 an 7 of the trajectory of Right Arm Abuction / Auction. Figure 2.13: The left plot shows the region where robot falls (/ oes not fall) an the right plot shows the variation in CM_Distance on varying the Points 6 an 7 of the trajectory of Torso Roll (angle name Torso 1). Figure 2.14: The left plot shows the region where robot falls (/ oes not fall) an the right plot shows the variation in CM_Distance on varying the Points 6 an 7 of the trajectory of Torso Pitch ( angle name Torso 2). 16 7/2/2007

17 Figure 2.15: The CM Distance on varying Point 3 of the trajectory of Right Leg Abuction/ Auction (angle name right leg 2) (left) an Point 4 of the trajectory of Right Leg Rotation (angle name right leg 3) (right) with the re line(bottom) showing the region in which robot falls / oes not fall. The range for which the fall is zero is the range in which robot oes not fall. In the next section we will see whether it is necessary to attain a certain amount of torso spee for sitting or not Effect of spee of torso If we see the variation of actual torso spee an vertical torso spee in trajectory plots of Figure 2.6, the torso spee starts builing up from the start of orange line. There coul be a possibility that if a minimum amount of torso spee it achieve at the start of the critical perio, the robot will always sit. If this is the case then we can easily preict whether robot will sit or not at the start of the critical perio. To stuy the effect of spee of torso on the sitting of robot, I varie the time of moving right arm (Flexion /Extension) DOF from just 1 TimeStep (64 ms) to 60 Time Steps. This time is 5 Time Steps in the original trajectory. By increasing the time of moving right arm (Flexion / Extension) DOF, torso spee can be ecrease. Therefore we can stuy whether a minimum amount of torso spee is require to make robot sit or not. Figure 2.16 shows the torso spee profiles on varying the time of moving right arm (Flexion / Extension) DOF over (1, 2, 5, 10, 15, 30 an 60) Time Steps. Also for comparison, the torso spee profile for a trajectory when robot falls has been plotte in re color. The robot was able to sit even when the time of moving right arm was 60 Time Steps. As can be seen in the plot, when the time of moving right arm is 60 Time Steps, torso spee starts builing up from zero spee. Therefore we can infer that we nee not achieve a minimum amount of torso spee for sitting. 17 7/2/2007

18 Spee (in m/s) Robot Falls Time (in sec) Figure 2.16: Torso Spee Profiles for ifferent time steps of moving right_arm_1 in critical region. The re plot (bottom most) is the one when robot falls. Torso_1 is set to 0.16 raians for this plot Conclusion of analysis From the analysis of the han-mae trajectory, following points can be note: (i) The sitting transition is ivie into 2 phases. (ii) The robot is unstable in the secon phase. (iii) The trajectory can be qualifie as goo or ba on the basis of its robustness. (iv) From the position of center of mass we are not able to istinguish between the trajectories which make robot sit an which make robot fall. This means that uring sitting robot always becomes unstable. Also there is not much variation in stability on the basis of position of center of mass. (v) The torso spee cannot be use to preict the sitting of the robot. Infact, it was notice that the trajectories in which arm oes not loose the contact of the groun in the critical perio, robot was able to sit. 18 7/2/2007

19 Overall we can say that robustness is more important than the stability of the sitting transition an some amount of instability is require to make the robot sit. The robustness has been achieve in the current han-mae trajectory. After the han mae trajectory has been analyze an optimize, I will escribe how we can make a ynamical controller for the trajectory. Chapter 3 Dynamical System for Sitting 3.1 Nee of Dynamical System an Main challenges The controller base on ynamical system is avantageous because the trajectories can be easily moulate by controlling only a few parameters of ynamical system. In case of eviation from the trajectory, system smoothly returns to the trajectory an aition of sensory feeback can also be one very easily using the stability properties of the system. To esign the ynamical system for the controller of crawling to sitting transitions, two major problems nee to be hanle that are: 1) Designing mathematical equations for sitting trajectories 2) Switching from crawling to sitting when a signal S is given By solving the first problem, we woul be able to moulate sitting transition by using only few parameters an by solving the secon problem we woul be able to make robot sit whenever we wish to, by controlling a signal S. In the next sections, I will escribe the ynamical systems for the above two problems. 3.2 Mathematical moel for sitting trajectories The sitting transition is ivie into two phases as explaine above. Therefore, first we create a ynamical system for the first phase. If we see the plots of variation of joint angles (Fig. 2.5), the egrees of freeom that move in the first phase are: Torso Pitch (Angle name: Torso_2) Right Leg Abuc / Auc (Angle name: Right_leg_2) Right Leg Rotation (Angle name: Right_leg_3) Left Knee (Angle name: Left_Knee) Left Leg Abuc / Auc (Angle name: Left_leg_2) Left Leg Rotation (Angle name: Left_leg_3) Also the egrees of freeom like: Right Arm Flex/Ext (Angle name: Right_Arm_1) Right Arm Abuc /Auc (Angle name: Right_Arm_2) Right Elbow (Angle name: Right_elbow) 19 7/2/2007

20 Left Arm Flex / Ext (Angle name : Left_arm_1) Left Arm Abuc /Auc (Angle name: Left_arm_2) Left Elbow (Angle name: Left_elbow) Right Leg Flex / Ext (Angle name: Right_leg_1) Left Leg Flex /Ext (Angle name; Left_leg_1) that were moving while crawling become constant while sitting. The equation for the egrees of freeom that were moving while crawling using x to enote the value of angle name can be written as follows: = y x y = α ( x x0 ) + β y Eq(1) where can be right_arm_1, right_arm_2, right_elbow, left_arm_1, left_arm_2, left_elbow, right_leg_1, left_leg_1 α x0 an β are constants whose values are same for all the DOFs is the constant value of the angle name ' ' uring first phase Among the egrees of freeom that start moving uring the first phase of sitting, first Torso Pitch starts moving as escribe in section 2.2. The equation for the movement of torso pitch using to enote the value of angle torso_2, can be written as follows: x x y β α = y = α = 160 = β ( x * β x0 / 32 ) + β y Eq(2) With this equation the angle torso_2 will move towars x0 which is the behavior we wish to have. The spee of the movement can be controlle by α an β. Then the movement of right leg an left leg is starte. The equation for the movement of right leg an left knee can be written in the same way as for torso pitch with an aitional factor that ensures that right leg an left knee start moving only after torso has starte moving. The equations are as follows: x = T ( y y = T ( α ( x T = 1+ e ) 1 x0 ) + β y 100 ( x 0.8 x0 ) ) Eq(3) 20 7/2/2007

21 where α α α right _ leg _ 2 right _ leg _ 3 left _ knee can be right _ leg _ 2, right _ leg _ 3, left _ knee = α = α = α / 2 / 2 β right _ leg _ 2 β β right _ leg _ 3 left _ knee = β = β = β / 2 / 2 / 2 x0 The value of T will be very small until the value of angle torso_2 is not 0.8*. Thus right leg an left knee will not move until the value of angle torso_2 is 0.8* x0. This ensures that when right leg an left knee start moving, weight of the boy is supporte mostly by right half of the boy. The T can also be multiplie by an external sensory signal which makes T approach 1 only when the boy weight is supporte by right half of the boy. Also as explaine in section 2.2, the movement of left leg (Abuc/Auc an Rotation) has to be synchronize with the movement of left knee. This is one using the following equations: x = T ( K c1 x where T is the same as in Eq(2) 1 K = 100 y left _ knee y left _ knee 1+ e can be left _ leg _ 3, left _ leg _ 2 c c 2 1 = β = 8 left _ knee + /10 c 2 ( x left _ knee x0 left _ knee ) ( x x0 )) Eq(4) When the left knee is extening, value of K is very small. Therefore the secon half of the equation will make x ecrease until either left knee stops extening or x becomes equal to x0. When the extension of left knee will complete, secon half of the equation will become zero as x left _ knee will become equal to x 0left _ knee, first half will become active as y left _ knee will become zero an will move x to zero. This completes the first phase of the sitting transition. The egrees of freeom that move in the secon phase are: Torso Pitch (Angle name: torso_2) Torso Roll (Angle name: torso_1) Right Arm Flex. / Ext. (Angle name: right_arm_1) Right Arm Abuc / Auc (Angle name: right_arm_2) The egree of freeom Torso Roll was not moving uring crawling or uring the first phase of sitting. Therefore its equation can be written as: 21 7/2/2007

22 x torso _1 y torso _1 α rtorso _1 e P = 1+ e P1 = 1+ e = P ( y = P ( α = α 1000*( P1 0.2) torso _1 torso _1 / *( P1 0.2) 1 ) ( x β torso _1 torso _1 100 ( x left _ leg _ 3 ) ( x left _ leg _ 3 ) x0 = β torso _1 1+ e ) + β / 4 torso _1 y torso _ ( x left _ knee x0 left _ knee ) ( x left _ knee x0 left _ knee ) ) Eq(5) The value of P will be very small uring the first phase of sitting an will become 1 when the first phase will en i.e when left leg (Rotation) becomes zero an extension of left knee is complete. Like variable T, variable P can also be multiplie by external sensory signal by which we can control the start of the critical phase of sitting. The DOF Torso Roll will start moving only when the first phase will en an P will become 1. The equations of egrees of freeom Torso Pitch, Right Arm Flex/Ext an Right Arm Abuc/Auc that were moving before also can be moifie by replacing x0 with ( x0 + P x1 x0 )). ( This will make these egrees of freeom move towars x1 when the secon phase of sitting transition will start. Similarly, when sitting is over i.e. hips touch the groun, we can change the attractor of left arm (Abuction / Auction) an right leg (Rotation) to make the sitting pose stable. The trajectories for relevant DOFs obtaine using above equations are shown in figure 3.1. As before, the robot starts sitting after yellow line (secon vertical line from the left), the orange line (thir vertical line from left) marks the en of first phase i.e P = 1 at this point. The re line (right most vertical line) is rawn when robot sits. Before the yellow line (secon vertical line from the left), robot is crawling an the black line (leftmost vertical line) is rawn at the time when signal for sitting is sent. The switching from crawling to sitting has been explaine in the next section. From the plots we can notice, that the trajectories similar to han mae trajectories have been obtaine until the re line (right most vertical line). The trajectory after this line only helps the robot sit in a proper position. So the trajectories nee not be same as han mae trajectory after re line (right most vertical line). 22 7/2/2007

23 Angle (raians)/ Spee (m/s) Angle (raians)/ Spee (m/s) Time (in sec) Time (in sec) Angle (raians)/ Spee (m/s) Time (in sec) 23 7/2/2007

24 Angle (raians)/ Spee (m/s) Time (in sec) Angle (raians)/ Spee (m/s) Time (in sec) Figure 3.1: Variation of joint angles of relevant egrees of freeom an torso spee (vertical in soli black line an absolute in otte black line) with time. (Top ) Variation of angles Right_arm_1 an Right_arm_2. (Secon from Top) Variation of angles Left_arm_1 an Left_arm_2.. (Thir from Top) Variation of angles Torso_1 an Torso_2. (Fourth from Top) Variation of angles Right_leg_1, Right_leg_2, Right_leg_3 an Right_knee. (Bottom) Variation of angles Left_leg_1, Left_leg_2, Left_leg_3 an Left_knee. Till black line (leftmost vertical line) robot is crawling. Then it gets signal for sitting at black line. At yellow vertical line (secon from left) robot starts sitting by moving its torso first. The orange vertical line(secon from left) is rawn at the time when robot has brought its left leg forwar an starts moving its right arm. The re line(right most) is rawn at the time when robot sits on the groun. The names of the angles whose variations are plotte are efine in Figure 1.2. In the next section I will escribe how we can smoothly switch from crawling to sitting. 24 7/2/2007

25 3.3 Switching from crawling to sitting While crawling, infants have a trot like gait, which is a perioic gait. The equations of the CPG that generates the trajectories for the hip an shouler joints for crawling as escribe in [2] are: xi= yi yi= α i yi ( K i K = ( e α = ν (1 + βe i i by i 2 ( μ x k s tan ce ky j + 1)( e σ x 2 i ) 2 i ) y ) K x 2 i k + + 1) ( e swing by i i i + 1) c 1 y j + c 2 y k Eq(6) where i = 1 4 enotes the ith oscillator, j the opposite oscillator an k the iagonal oscillator, c1 an c2 are positive coupling constants for the CPG architecture shown in figure 3.2 below. Figure 3.2: The architecture of CPG (Source [2]) We want when we switch S from 0 to 1, we shoul get sitting controller. For this, we can get a combine signal from the oscillator as well as from the ynamical system for sitting an use the signal S to shut off the oscillator or the ynamical system for sitting by moifying their equations as shown below: For oscillator = (1 S) f 1 ox = (1 S) f 2 oy Combine Signal ( ox ( ox, oy, oy ) ) cx = ox + sx For DynamicalSystem = sx = sy Eq(7) S f S f 3 4 ( sx ( sx, sy, sy The problem with this equation is that the controller will shift abruptly from crawling to sitting on switching S from 0 to 1. By abruptness I mean that even if the leg is moving backwar, it will immeiately start moving forwars without completing its cycle as shown in Figure 3.3. This will require large amount of acceleration. Therefore we wish ) ) 25 7/2/2007

26 that after the signal S is change from 0 to 1, switching from crawling to sitting controller shoul occur only when while crawling hip an shouler joints are moving in the same irection as they will move after shifting from crawling to sitting. During sitting, hip an shouler joints are extene to large extent, therefore it is esirable that switching from crawling to sitting takes place when hip an shouler joints start extening. This can be achieve by substituting S by ' efine as: Si ' = S e ' i D osy i S i S will become 1 only when S is 1 an osyi is positive i.e osxi is increasing. For the egrees of freeom relate to Left Arm, S will be substitute by S 1 ', for Right Arm by S ' 2, for Right Leg by S 3 ' an for left leg by S 4 '. For Torso, S is substitute by S 3 ' as movement of torso results in increase pressure on right leg. The effect of this substitution is shown in the figure 3.3 below. Also in Fig. 3.1, the black line (left most vertical line) correspons to signal S an yellow line (secon vertical line from left) correspons to signal S 3 '. Figure 3.4 shows snapshots of the robot switching from crawling to sitting when following the trajectories shown in Fig 3.1. Figure 3.3: Switching from crawling to sitting. Effect of replacing switching signal S by S 26 7/2/2007

27 Figure 3.4: Robot switching from crawling to sitting. In the first 3 snapshots, robot is crawling. The signal for sitting is given at the time of thir snapshot but torso starts moving only by 6 th snapshot as right leg is going backwars in thir snapshot. Left leg an arms become constant after thir snapshot. The snapshots have been taken from vieo: Craling_2_sitting_right Thus smooth switching from crawling to sitting oscillator is achieve. Conclusion an Future Work A controller for sitting of the robot in the same way as infants has been successfully implemente. The sensory feeback can easily be integrate into this controller by moifying the values of variable T (efine in Eq(3)) an P (efine in Eq(5)) accoring to the sensory input. The robot can be switche from crawling to sitting smoothly at anytime by proviing an external signal S. Also the main characteristics of the sitting behavior of infants an the perio of instability have been ientifie. The future work that can be one is: 1) Aition of sensory feeback while sitting. This will be particularly useful when robot is in critical perio. It was observe that robot falls in the critical perio when arm looses contact of the groun. Sensory feeback might be very helpful in preventing the fall. 27 7/2/2007

28 2) Collection of biological ata to know that when infants enter the critical perio of sitting phase, their movements are controlle by signals from brain or signals from spinal cor. 3) Development of controller for transition from sitting to crawling 4) Increase in the limit up to which hip joints can be flexe / extene. References [1] G. Sanini, G. Metta, an D. Vernon, Robotcub: an open framework for research in emboie cognition, 2004, paper presente at the IEEE-RAS/RJS International Conference on Humanoi Robotics, Santa Monica, CA. [2] L. Righetti an A.J. Ijspeert. Design methoologies for central pattern generators: an application to crawling humanois, Proceeings of Robotics: Science an Systems 2006, Philaelphia, USA [3] Michel, O. Webots:Professional Mobile Robot Simulation. Int. J. of Avances Robotic Systems, 2004, pages:39-42,vol.1 [4] G. Metta, G. Sanini, D. Vernon, D. Calwell, N. Tsagarakis, R. Beira, J. Santos-Victor, A. Ijspeert, L. Righetti, G. Cappiello, G. Stellin, F. an Becchi. The RobotCub project - an open framework for research in emboie cognition, Humanois Workshop, Proceeings of the IEEE -RAS International Conference on Humanoi Robots, December 2005 [5] MATLAB Function pchip: Fritsch, F. N. an R. E. Carlson, "Monotone Piecewise Cubic Interpolation," SIAM J. Numerical Analysis, Vol. 17, 1980, pp /2/2007