A CONTROL ARCHITECTURE FOR DYNAMICALLY STABLE GAITS OF SMALL SIZE HUMANOID ROBOTS. Andrea Manni,1, Angelo di Noi and Giovanni Indiveri

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1 A CONTROL ARCHITECTURE FOR DYNAMICALLY STABLE GAITS OF SMALL SIZE HUMANOID ROBOTS Andrea Manni,, Angelo di Noi and Giovanni Indiveri Dipartimento di Ingegneria dell Innovazione, Università di Lecce, via per Monteroni, 7 Lecce, Italy, Fax: , {andrea.manni, giovanni.indiveri, angelo.dinoi}@unile.it Abstract: From the 97 s biped robots have had a large attention from the robotic research community. Yet the issue of controlling dynamically stable walking for arbitrary biped robots is still open. We propose a simple control architecture based on the use of the FRI (Foot Rotation Indicator) point and the support polygon. The major advantage of the proposed architecture is that motion planning (and eventually sensor based re-planning (slower feedback loop)) is limited to the leg joints whereas the trunk and arm degrees of freedom are controlled in closed loop (faster feedback loop) to achieve overall dynamic stability. Such architecture allows to decouple the problem of dynamic stable walking in the two relatively simpler problems of gait generation and robot stabilization. This architecture is particularly suited for small size robots having limited onboard computational power and limited sensor suits. The effectiveness of the proposed method has been validated through Matlab R simulations and experimental tests performed on a Robovie-MS platform. Copyright c IFAC Keywords: control architecture, dynamically stable gait, foot rotation indicator, support polygon.. INTRODUCTION Humanoid robots are enjoying increasing popularity as their anthropomorphic body allows the investigation of human-like motion and multimodal communication. Currently, examples of advanced humanoid robots include Asimo (Honda, Inc) or Qrio (Sony, Global), whereas simpler designs include Vstone (Vstone, Co. Ltd), Kondo (Kagaku, Co. Ltd) or RoboSapien (Wowwee, Robosapien), which has been developed for the toy market. Small humanoid robots can have from a few to Corresponding author over twenty actuated degrees of freedom and generally they carry some microcontroller electronics board for the low level control, i.e. to generate target signals for the actuators. Higher level control loops need by far more computational power and are usually implemented either on industrial controllers as, by example, PC or on handheld computers (Behnke et al., July a)(behnke et al., May b), such as Pocket PC and alike. The communication link between low and higher level control loops is most often of serial RS kind. The actuators are usually low power servo motors from the radio control (RC) models market.

2 Fig.. Robovie-MS Biped locomotion is one of the most important issues to be faced: the basic characteristics of all biped locomotion systems are (Vukobratović and Borovac, ): () the possibility of rotation of the overall system about one of the feet edges caused by strong disturbances; () gait repeatability; () regular interchangeability of single and double support phases. In this paper, we propose a control architecture for the motion control of a humanoid robot that allows to decouple the gait generation issue and the overall dynamic stability of the system. The analysis of dynamic stability is addressed on the basis of the Foot Rotation Indicator (FRI) (Goswami, 999). The remainder of the paper is organized as follows: the used robot is described in the section, the proposed architecture in section. Implementation issue are discussed in section. Simulation and experimental results are presented in section and, finally, conclusions are briefly discussed in section.. ROBOT DESCRIPTION The considered robot, shown in figure, is the Robovie-MS, a small humanoid robot kit made by Vstone (Vstone, Co. Ltd). The robot has 7 degrees of freedom (DOFs): in each leg, in each arm and one in the head. It is 8 cm tall and has a total weight of about 8g. It has one axis acceleration sensor and 7 joint angle sensors. The servos control board is composed by an H8 CPU at MHz, a KByte FLASH- ROM memory, a KByte RAM and a 8KByte External-EEPROM.. CONTROL ARCHITECTURE In this section we present the control architecture proposed to obtain a dynamically stable gait for the biped robot. The issue of planning desired Fig.. Definition of the support polygon (single (a) and double (b) support phases) joint trajectories for dynamic walking is an important research area: several methods have been presented in the literature. Some of these are based on the inverted pendulum model for the biped legs (Tsuji and Ohnishi, ), (Goswami et al., 997). Other more complicated techniques take directly into account dynamic stability indicators as the zero moment point (ZMP) (Zhou et al., ) or the foot rotation indicator (FRI) (Hoffman et al., ). We consider a control architecture based on the FRI (Foot rotation Indicator) (Goswami, 999), which is a point on the foot/ground contact surface where the net ground reaction force would have to act to keep the foot stationary. To ensure the absence of foot rotations around any axis laying in the ground plane, the FRI point must remain within the convex hull of the foot support area. The proposed control architecture is shown in figure. The basic idea, is that the leg joints only are considered for locomotion planning while the upper body and arm joints are used for dynamic stabilization. The aim is to design an architecture that allows to decouple the gait generation and dynamic stabilization problems. In essence, this architecture is inspired by classical task based control architectures (Sciavicco and Siciliano, ) of industrial robots that allow to design separate control laws for concurrent, but different tasks. As for the gait generation, basically this will be commanded to the leg joints based on an off line (or loose feedback) planning phase. Any of the standard gait planning approaches described in the literature may be considered, eventually including obstacle avoidance tasks as suggested in figure. With reference to figure, the gait generator is a planner for the leg joints only generating either leg joint torques (in case of a dynamic planner) or leg joint reference velocities (in case of a kinematics planner). In either case the gait generator output is used to define the leg joint commands. As pictorially represented in figure, such planner may use joint information to perform obstacle avoidance planning or re-planning. As for the

3 Fig.. Control architecture. dynamic stability control, the direct kinematics model will be used to compute the velocity and position of the center of the support polygon as a function of the joint values. The support polygon can be defined as the contact area between the humanoid robot and the ground; therefore in the single support phase, it is given by the sole in contact with the ground (seen figure (a)), while in the double support phase, it is given by the convex hull of contact points between the soles and the ground (seen figure (b)). The dynamic stabilization controller will have as input the vector difference of the position of a target point inside the support polygon with the position of the FRI, and its aim will be to drive this error to zero by acting on the upper body degrees of freedom. Indeed if FRI is in the support polygon, we are sure that the robot s motion is dynamically stable. It should be noticed that according to its definition (Goswami, 999), the FRI tends to the ground projection of the center of mass (GCoM, in the sequel) as the joint accelerations tend to zero. Namely indicating with r F RI and r GCoM the position of the FRI and of the ground projection of the center of mass respectively, the following holds: δ := r F RI r GCoM = lim a i, ω i δ = () being a i and ω i the linear and angular accelerations of each link. Notice that δ is a continuous function of the link accelerations and that if a j < g j, then δ is upper bounded. Decoupling the gait planning and dynamic stabilization tasks is particularly important for small size robots that have limited computational power. In order for such decoupling to be effective, the cycle time of the dynamic stabilization controller needs to suitably smaller than the gait period. Moreover the legs contribution to the FRI position will be regarded as a disturbance (namely a non manipulable input) by the upper body controller. Notice that this task division approach can be compared to (Khatib et al., ) with the difference that in (Khatib et al., ) the task division is implemented at an algorithmic level, while in the present case at a control architecture level.. IMPLEMENTATION ISSUES The dynamic stabilization feedback control loop described in section and in figure can be designed based upon a Lyapunov technique. Wanting the FRI point to converge on a target point r t within the support polygon, a quadratic Lyapunov candidate function may be defined as: V = (r t r F RI ) T R (r t r F RI ) () where R will be a symmetric positive defined matrix, and r F RI and r t are the positions of the FRI and of a target point inside the support polygon respectively. The time derivative of V will be given by: V = (ṙ t ṙ F RI ) T R (r t r F RI ) = () ( = ṙ GCoM ṙ t + δ ) T ( ) R r t r GCoM δ being δ defined in equation (). Calling q L, q UB the legs and upper body joint variables and J L (q), J UB (q) the legs and upper body Jacobian matrices such that ṙ GCoM = J L (q) q L + J UB (q) q UB, () equation () suggests to compute the reference value of the upper joint velocities as [ ] q UBd = J UB R (r t r GCoM ) J L q L + ṙ t () being J UB the Moore Penrose pseudo inverse of matrix J UB. Notice that the use of the ground

4 projection of the center of mass in place of the FRI in the control law makes V equal to V = (r t r GCoM ) R T R (r t r GCoM ) + + δ R δ ( δt δ T R T ) R (r t r GCoM ) that is negative definite only in the limit of vanishing δ and δ. Nevertheless the use of r GCoM in place of r F RI makes the control law computationally much simpler as according to the FRI definition (Goswami, 999), ṙ F RI will depend on the joint accelerations and jerks. As for the q L, q L and ṙ t in the upper body joint control law (), notice that q L and q L are known as they are generated by the legs gait controller and ṙ t is generated in such a way that r t is always within the support polygon. Generally ṙ t is designed such that during the single support phase r t moves within the support polygon in the same direction of the walking gait so that at the end of the single support phase, when the support polygon becomes the one of the double support phase, r t will be located approximately in its center. Considering only the sagittal plane, for the sake of simplicity, the robot model is represented in fig.. In this case the Jacobian matrices are : J L (q) = k s k s k s k s k c k c k c k c M and J UB (q) = = M k s k s k s k 7 s 7 k c + k c k c k 7 c 7 where M( is the total mass of the robot and k r m + m + m + m + m + m + ) ( m 7, k r m + m ) + m + m + m + m 7, ( k r m + m ) ( + m 7, k r m + m ), m k r, k m r, k m 7 r 7 7 and m i and r i are respectively the mass and the length of the i-th link as reported in Table.. SIMULATION AND EXPERIMENTAL RESULTS The presented control approach has been validated in the sagittal plane both in simulation and experimentally. The simulations have been performed in Matlab using the Robotics Toolbox realized by P. I. Corke (Corke, 99). The experimental implementation has been realized using The notations s i...j, c i...j represent, respectively, sin(q i q j ), cos(q i q j ) Fig.. Robot model in sagittal plane the Robovie-MS platform. In figure simulation results are displayed relative to a case where the robot executes a complete step in the positive x axis direction. Initially the two feet have a common x coordinate and the target point r t is at the center of the (initial) double support phase support polygon. While one of the two feet stays on the ground, the other one moves forward giving rise to the single support phase. During this phase, the target point velocity ṙ t is constant along the positive x axis direction until r t reaches the border of the single support phase support polygon where ṙ t is set to zero. As expected, figure () shows that the error r t r GCoM is driven to zero by the action of the upper body joints (torso and arms). The parameters used for simulations are reported in Table. N Link Length (cm) Mass (g) 9,,88,9 7,88 9,,. 7,. Table. Robovie-MS parameters used for the simulations. Experimental results relative to the implementation of the proposed control law for the sagittal plane of the Robovie-MS platform are reported in figure (). Notice that, during the experimental validation tests, the robot needs to perform a dorsal plane movement (figure () (d)) to prevent the GCoM point from falling out of the single support phase support polygon in the z direction. Such dorsal plane motion is commanded in open loop. Moreover, as the joints are actuated by position servo motors, the control law () has been integrated in order to compute position commands for the upper body joints. Given the limited communication bandwidth with the joint position servo controllers, during the experimental tests the position commands (labeled as viapoints in figure ()) were updated at very low frequency (approximately Hz). Nevertheless, as shown in

5 figure (), the sagittal plane component of the GCoM point converges to its target value. In particular, the experimental results reported in figure () refer to a situation where the robot starts from a double support phase: the right foot is moved upwards giving rise to a single support (on the left foot) phase during which the left knee is bend contributing to move the GCoM in the negative x direction. Such destabilizing x axis motion of the GCoM is automatically compensated for, in the sagittal plane, by the upper body joints controlled by the proposed law. Overall, the x coordinate of the GCoM converges to its constant target position approximately located at the center of the support polygon.. CONCLUSION A humanoid robot control architecture has been presented that allows to decouple the gait generation issue from the dynamic stabilization one. To the best of the author s knowledge, this approach appears to be novel. The proposed solution is particularly suited for small size, low cost humanoid systems having limited computational power. Although due to HW constraints the experimental validation was possible only at rather low control update frequencies (approximately Hz), extensive trials have shown that leg motions that would have caused the robot to fall in case the upper body joints were kept still did not cause the robot to fall when the upper body joints were controlled according to the presented strategy. Simulations performed at higher update frequencies and higher link velocities confirm the effectiveness of the presented solution. Several open issues need to be addressed in future work. These include handling of: (i) dynamic effects by estimating δ := r F RI r GCoM and its time derivative, (ii) possible link collisions and (iii) jacobian singularities. A possible strategy to handle link collisions and jacobian singularities is to replace the pseudo-inverse of J UB in equation () with a weighted pseudo-inverse such as ( ) W JUB T JUB W JUB T for some symmetric positive definite W of proper dimension. The extra degrees of freedom provided by W can be used to try avoiding collisions and kinematics singularities. st International Conference on Dextrous Autonomous Robots and Humanoids (darh) Yverdon-les-Bains - Switzerland. May. Corke, P.I. (99). A robotics toolbox for MAT- LAB. IEEE Robotics and Automation Magazine (),. Goswami, A. (999). Postural stability of biped robots and the foot-rotation indicator (fri) point. The International Journal of Robotic Research Vol. 8, No.,. Goswami, A., B. Espiau and A. Keramane (997). Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Autonomous Robots, 7 8. Hoffman, A., M. Popovic, S. Massaquoi and H. Herr (). A sliding controller for bipedal balancing using integrated movement of contact and non-contact limbs. in Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and System, (Sendal, Japan). Honda (Inc). Kagaku (Co. Ltd). Khatib, O., L. Sentis, J. Park and J. Warren (). Whole-body dynamic behavior and control of human-like robots. International Journal of Humanoid Robotics Vol., No., 9. Sciavicco, L. and B. Siciliano (). Modelling and Control of Robot Manipulators. Springer. Berlin. Sony (Global). Tsuji, T. and K. Ohnishi (). A control of biped robot which applies inverted pendolum mode with virtual supporting point. AMC, Maribor, Slovenia. Vstone (Co. Ltd). Vukobratović, M. and B. Borovac (). Zeromoment point - thirty five years of its life. International Journal of Humanoid Robotics Vol., No., 7 7. Wowwee (Robosapien). Zhou, C., P. K. Yue, J. Ni and S. B. Chan (). Dynamically stable gait planning for a humanoid robot to climb sloping surface. Proceedings of the IEEE Conference on Robotics, Automation and Mechatronics, Singapore. REFERENCES Behnke, S., J. Muller and M. Schreiber (July a). Playing soccer with robosapien. In: In Proceedings of The 9th RoboCup International Symposium, Osaka, Japan. July. Behnke, S., J. Muller and M. Schreiber (May b). Using handheld computers to control humanoid robots. In: In Proceedings of

6 . Upper body joints (deg) x coordinate of GCoM (cm)... θ θ θ7. x coordinate of Ground projection of Center of Mass Pos. target point. time (sec) (a) 8 7 time (sec) (b) 7 y (cm) Upper body joint velocities (deg/s) dθ/dt dθ/dt dθ7/dt time (sec) (c) 7 x (cm) (d) Fig.. Simulation : (a) position of the x coordinate of GCoM (solid line) and position of target point (dashed line); (b) θ, θ and θ7 position, (c) θ, θ and θ7 velocity, (d) stick diagram of start (solid line), middle (dashed line) and final (dotted line) configuration and position of the center of the support polygon ( ) and the position of the center of mass ( ) Fig.. Experimental results : (a) position of the x coordinate of GCoM (*) and position of target point (dashed line); (b) Robovie-MS initial position, (c) Robovie-MS middle position, (d) Robovie-MS final position