A Compact Model for the Compliant Humanoid Robot COMAN

Transcription

1 The Fourth IEEE RAS/EMBS International Conference on Biomedical Robotics and Biomechatronics Roma, Italy. June 24-27, 212 A Compact for the Compliant Humanoid Robot COMAN Luca Colasanto, Nikos G. Tsagarakis, and Darwin G. Caldwell Abstract This work reports on the modelling of a compliant humanoid robot. The introduction of intrinsic compliance in some of the joints of the COmpliant humanoid robot (COMAN) affects its dynamics making no trivial the generation of walking gaits. To permit the development of effective gait generators which take into account these additional dynamic effects due to intrinsic compliance, an appropriate model which can predict the robot motion dynamics is required. To predict the motion of the centre of mass (COM) of the compliant robot we propose a new model for compliant humanoid systems which combines the inverted pendulum model approach with a three dimensional compliant model (Cartesian) at the level of the COM. The derivation of the model is introduced followed by experimental validation which demonstrates the adequate performance achieved by the proposed reduced model. In particular the efficacy of the model was experimentally validated on the COMAN humanoid platform using a series of ZMP based walking gaits. M I. INTRODUCTION ost of the existing state of the art humanoid robots typically employ stiff actuation technologies to power their bodies. [1-11]. Looking on the details of the actuation system of these highly complexity machines one can notice that the predominant approach is the use of nonbackdrivable, stiff transmission systems, combined with high gain PID controllers. This highly non-backdrivable and large stiffness approach has been mostly inspired from industrial robotics designs where the necessity for precision movement and high load disturbance rejection is predominant. Although this industrial approach gives to the humanoid robots high precision, the resulting large output mechanical impedance makes these inherently unsafe when interacting with humans or environment. Safety is not the only feature being compromised in these stiff humanoids. Despite the fact these stiff humanoid designs represent outstanding engineering prototypes, when compared to biological systems they have significant performance handicaps relating particularly to their energy efficiency, peak power level and ability to adapt to interaction uncertainties. To permit the widespread of humanoid systems and allow the development of applications within the domestic human living environment humanoid body systems should exhibit characteristics such as lightweight structures, high mobility and large adaptability to interaction uncertainly which may Luca Colasanto (luca.colasanto@iit.it), Nikos G. Tsagarakis (nikos.tsagarakis@iit.it), Darwin G. Caldwell (darwin.caldwell@iit.it) are with the Department of Advanced Robotics, Istituto Italiano di Tecnologia (IIT), Via Morego 3, Genova 16163, Italy occur with humans or environment. Nowadays a multidisciplinary effort is dedicated in that direction and gradually humanoid robotics design is taking distance from the traditional industrial robotics approach offering its own solutions to solve problems. The so called biomimetics is probably the main principle followed by the research community of this field and it refers to a novel way to design hardware and develop control strategies inspired by the biological systems. COMAN robot, Fig. 1, is a bipedal robot which follows this approach by implementing inside the actuation system the physical principle of compliance [12]. In particular a passive elastic mechanism is inserted in the joint between the electrical motor and the link [13]. Other examples of humanoid robots with passive elastic elements in the actuation system includes the work in [14] which introduced compliance into the robot joints using two motors in antagonistic configuration and non linear springs and the work of [15] which uses linear series elastic actuators to power the lower limb of a humanoid robot. Fig.1 The mechanical assembly of the COMAN lower body. This elastic transmission adds an extra effect in the robot dynamic behaviour which is not present in stiff robot. The dynamic behaviour is different from that of the stiff joint as the link angle always differs from the motor angle according to the joint torque. This introduced joint error and deterioration of the tracking performance may significantly affect the humanoid robot stability. The main paradigm of dynamic walking is to ensure the equilibrium between reaction force of the ground, gravitational force and inertia /12/$ IEEE 688

2 force. The compliance reduces the effect of take off and touchdown disturbance of the foot. In fact instead of a stiff response to the disturbance it introduces a proportional action to the position error (an intrinsic proportional controller). However, the same time compliance causes a displacement between the foot position reference and the real position of the foot also during the rest of the walking cycle of the robot which may violate the stability criterion such as ZMP [16]. If the stability constraints of the trajectory developed are enough conservative and far from instability condition (for example in ZMP based trajectory it means ZMP position far from edges of the foot sole) the stability of the walking can be in generally preserved otherwise the extra dynamic introduced by the compliance and disturb the walking gait significantly will eventually cause the robot to fall down. Therefore, while ZMP based gait trajectories have been successfully applied to stiff humanoid where high precision tracking performance is possible their application to compliant humanoid systems is not trivial due to the deterioration of the joint tracking performance [17]. To permit the application of these trajectory generation techniques also to compliant systems as well to develop control schemes for these compliant robots appropriate models which can effectively predict the dynamic behaviour of these highly complexity compliant humanoids are required. The aim of this work is to develop a reduced model able to approximate the behaviour of the COMAN robot. The presentation of the work is organized as follows: Section II introduces some details of the compliant humanoid COMAN. Section III reports on the system modelling starting from models of the compliant transmission in a single joint and progressively presenting the equivalent compliant Cartesian models at the level of the COM which are finally combined with a inverted pendulum based model to realize the proposed reduced model. The tuning and experimental validation of the model is introduced in section IV while finally, section V addresses the conclusions. II. COMPLIANT HUMANOID HARDWARE The COMAN humanoid robot, Fig. 1, is being developed within the AMARSI European project, which aims to achieve a qualitative jump toward rich motor behaviour in robotic systems, rigorously following a systematic approach in which novel mechanical systems with passive compliance, control, and learning solutions will be integrated. With regards to the mechanical systems with passive compliance the goal is [12]: - to reduce the distinction between plant and controller, that is typical in traditional control engineering, to fully exploit complex body properties, - to simplify perception, control, and learning, and to explore how compliance can be exploited for safer human robot interaction, reduced energy consumption, simplified control, and faster and more aggressive learning. The mechanical structure of a leg of the COMAN humanoid and an overview of its kinematics with the location of the DOF are illustrated in Figure 2. From the kinematic perspective the new lower body includes the lower torso (housing the 3 DOF waist module), and the two leg assemblies with 6 DOF each. The height of the COMAN lower body, from the foot to the waist, is 671 mm, with a maximum width and depth (at the hips) of 176 mm and 11 mm, respectively. The total lower body weight is 17.3 kg, with each leg weighing approximately 5.9 kg, and the waist section, including the hip flexion motors, weighing 5.5 kg. Each joint is equipped with three position sensors and one torque sensor. This permits to measure the position of each joint before and after the elastic transmission. In addition, two six axis force/torque sensors are mounted below the ankle in order to measure the interaction forces with the ground. Fig.2 COMAN leg assembly and kinematics. The leg of COMAN incorporates two series elastic (SEA) actuation units, which are placed at the knee flexion and the ankle dorsiflexion joints. The SEA actuation units used in the COMAN humanoid robot are based on a compliant actuation unit developed in [13]. III. COMPLIANT HUMANOID MODELING The incorporation of passive compliance into the joints of COMAN humanoid results in a multi-dof nonlinear spring mass system. To permit the effective control of the system a model which can accurately describe the system behaviour is required. Following a bottom-up approach the joint models/characteristic are initially identified and used to obtained a Cartesian spring mass damper model at the level of the centre of mass (COM). Finally the model equation are defined, simplified and then express in a compact representation. 689

3 A. Actuation and Joint modeling In COMAN robot two types of joints exist. The first type is a stiff joint actuated by a motor and a harmonic reduction drive group. The passive compliance in these joints is due to the elasticity of the harmonic gearbox drive ( ). The actuation of the second type of joint which is called compliant and used in knee and ankle flexion incorporates an additional physical elasticity which was realized based on the actuation unit presented in [13]. The additional elastic mechanism is in series with the harmonic drive and is characterized by stiffness. Fig. 3 Joint model. Fig. 3 present a schematic model of the joint. According to this representation and by adapting the model form of [18] the joint can be described by the following equations: (1) = (2) where, and are the position, velocity and torque of the motor respectively reflected at the link side after the gear reduction: (3) where N is the gear ratio (N=1:1), and are position and torque of the motor. and are inertia and damping of the motor reflected to the link side as follow: (4) (5) (6) where and are the torque sensitivity and back EMF constant, is the stator resistance and is the physical damping of the motor. Finally,,,, and are the position, velocity, torque, inertia and damping of the link respectively and is the resultant joint stiffness ( for the stiff joint and for the case of a compliant joint). In COMAN robot the motor position is measured by a 12bit incremental encoder and on the link side and are measured by a 12bit absolute encoder and a strain gauge based custom torque sensor. In the case of the compliant joint it is possible to approximate the resultant joint stiffness to since is much larger than. To obtain the stiffness/damping parameters of the joint of the COMAN robot perturbation experiments were performed. The first set of experiments was carried out to compute the stiffness value of joints. The motor was controlled to a fixed position ( ) and an increasing torque (measured by the torque sensor) was quasi statically applied to the joint (in order to eliminate the dynamics effect). The deflection of the link position with respect to the harmonic drive output position was measured by the encoders. The relationship to identify is: where and are the measured joint torque and link deflection and represent the torque measurement offset. Using least squares approximation techniques the value of was computed. The same procedure was repeated for each joint. The second set of experiments was executed to compute the physical damping of the joints. To estimate this value the robot was controlled to keep a standing posture. The motor position was fixed to a constant value ( ). Under these conditions the link positions in the knee and ankle joints can still be deflected since they are elastically coupled to the output of the gearbox which was stiff position controlled. External perturbations were applied to the standing robot and the induced oscillations were recorded through the proprioceptive sensors of the robot. For decoupling the knee and the ankle the elasticity of one of the two joints was locked at the time. By monitoring the decaying oscillations the damping presented at the joints was estimated by approximating the system with an inverted pendulum model. Having identified the stiffness and damping of each joint the joint stiffness and damping matrices can be defined for each leg respectively: Both of them are 6x6 diagonal matrices positive defined,,, i={1,6} B. Centre of Mass (COM) Equivalent Cartesian model To generate effective trajectories for the compliant humanoid a model of the overall system which considers the joint elasticity property is required. In this work, we examine the validity, as an adequate model to represent the compliant robot dynamics, of an equivalent spring-mass Cartesian model at the level of the centre of mass (COM). For each leg the Jacobian matrix from the foot base frame placed below the ankle, to the frame placed at the COM of the robot has been computed (Fig. 4). The relationship between the Cartesian velocity of the COM and angular velocity of the leg links is: where is the vector of link joint velocities and are the linear velocity and angular velocities of the COM respectively. In addition the Cartesian force/moments at the COM are linked with the joint torques through the following expression: where represents the external forces and momentum acting at the level of the COM, is the 6x1 vector of joint torques needed to balance the external forces and s the transpose of the Jacobian (7) (8) (9) 69

4 matrix. The torque developed to the joints can also be expressed as a function of the deflection of the compliant transmission: (1) Similarly, the relationship between displacement of the COM frame and forces developed due to this displacement is given by: (11) where is the Cartesian Stiffness matrix which maps the effect of the joint compliance into Cartesian space (at the COM) of the robot. Equation (11) is an approximation of the full relationship because it doesn t take into account the change of the Jacobian matrix during the movement [19]. This approximation is valid if the deflection is small: this condition occurs during the model identification experiments as reported in the following sections. By some manipulation of (8), (9), (1) and (11) the Cartesian stiffness matrix can be obtained as a function of the joint stiffness matrix : (12) where is the inverse of transpose of the Jacobian matrix. In a similar manner, the Cartesian damping matrix at the pelvis level can be obtained from the diagonal joint damping matrix as follows: (13) The Cartesian damping matrix, maps the effect of the physical damping of the joint space into the Cartesian space of the pelvis frame. C. Overall model principles The model of lower body COMAN robot dynamics is developed with the following considerations in mind: (A1) The joints positions before the elastic transmission are controlled with a stiff PID loop. (A2) The elasticity in the joint transmission system is due to the harmonic drive compliance as well due to additional physical elasticity integrated in some of the pitch joints of the leg (knee and ankle), see section II. (A3) A single mass approximation is used for the robot model. Due to the first assumption it is possible to reduce the complexity of the model. In the case of an ideally stiff position control (motor position error equal to zero) combined with high reduction ration (minimum backdrivability) the dynamic of the motor in (1) can be ignored when the robot is subject to external force perturbations. In this case, (2) approximate the overall joint/link dynamic because the dynamics of the controlled actuator is much faster than the dynamics of the transmission. The consequence of (A2) is that the level of compliance is high in sagittal plane of the humanoid robot (due to additional elasticity in the knee and ankle pitch joints) while in lateral direction the robot is stiffer (only the compliance of the harmonic reduction drive contributes to this). Experiment results confirm the above by demonstrating large deviations of the pelvis (COM) position during the walk along the sagittal and vertical directions x and z and smaller deviations along the lateral direction y according to foot frame shown in Fig.4. Because of that, in y direction the movement can be approximated by a stiff system. Let now considered the forces generated at the pelvis (COM) frame when the COM position is deflected from its reference position vector to a position. Using (1) and (11) the generated forces can be related to the COM deflection as follows: (14) where are sub matrices of related to the linear motion along x, y and z. In case of diagonal matrices x, y and z dynamic are completely decoupled, however, this is not the case for the matrices in (12) and (13) in which the off diagonal elements are different from zero. It means that decoupling the movement of the robot is no possible. Fig. 4 Robot model, COM and associated support feet reference frame. As all humanoid COMAN is a distributed mass system. In this work we though consider a single mass approximation model. This is a common approach which has been extensively used in trajectory generation and control of humanoid robot [2]. Our interest is to validate if this is also applicable in the case of a humanoid in which additional complexity due to the elasticity of the joints is present. Therefore, according to (A3) the dynamic of the robot of the robot is approximated to the dynamic of the single mass placed at the pelvis (COM position). Due to the intrinsic joint elasticity and relevant passive damping the linear passive dynamics of the single mass model can be described by the following expression: (15) where with m being the total mass of the robot placed at the COM, is the Cartesian forces given by (14) and represents the gravity. By 691

5 omitting the passive dynamics along y (lateral direction, physical elasticity is not present in the joints contributing to that direction) and considering only the passive dynamics along the sagittal and vertical directions (15) can be written in a matrix form as follows (16) where represent the forces along x and z directions due to the equivalent Cartesian stiffness and damping. Considering (14), (16) can be further extended as follows: (17) (21) where,, and, i={1,2} are parameters inserted to compensate for model errors due to the approximation used as well other errors from the identification of the joint stiffness and damping parameters. The implementation of the system in (2) and (21) is shown in Fig. 6: where,, and are the relevant elements of,,,, are the relevant elements of, end are Cartesian position reference of the COM, and,,,, are position, velocity and acceleration of the COM when it is subject to external loads. The passive dynamic of the COM of the robot in Cartesian space during stance phase is described by (17). IV. MODEL VALIDATION The efficacy of the introduced model was experimentally evaluated through by executing walking gaits and comparing the motions of the COM of the experimental platform with those predicted by the simulation of the model. The reference trajectories for the robot were generated based on the ZMP approach. During the single support the robot was approximated with a single mass linear inverted pendulum. The ZMP trajectory was computed in order to achieve the desired gait considering the step length (sl), single support duration ( ), double support duration ( ) and the feet distance in double support. Using the linear inverted pendulum model [2], [21] the reference position of COM can be obtained from the defined ZMP reference. (18) (19) In the above represents the fixed COM height. The reference trajectories of the joints were then computed from the COM position through inverse kinematic. A. Implementation The model in (17) can be rearranged as follows: (2) Fig. 6 block diagram Equations (2) and (21) describe the system during single support. In fact during this phase the robot stands on a single leg with the compliant joints of the support leg to mostly affecting the robot movement. During the double support both legs are on the ground. In this configuration all compliant joints from both legs affect the robot dynamics. To take into account the effect of the second leg during the double support phase the same analytical procedure used to define equation (2) and (21) can be iterated for the second leg in order to derive the forces at the COM along x and z as generated due to the deflection of the second leg. However to reduce the overall model complexity a different approach was followed in this work. During the double support phase it is assumed that the two feet on the ground do not move relative to each other. The forces developed from the two legs are different because of the different configuration of the legs and different displacements. It is possible though to include the effect of the second leg by scaling (2) and (21) as follows: (22) (23) Equations (22) and (23) are used during the double support phase. The scaling coefficients are computed from experimental data which evaluates the x and z forces measured by the force/torque sensors installed at the feet of the robot. In Figure 9 the measured contact forces along the vertical direction z during a single walking cycle are presented. The 692

6 Z [m] X [m] Z [m] X [m] Z force [N] red curve is the average value of the right leg vertical force component during the single support and the two double support phases. The force along z direction measured by the sensor of the left leg is represented by the dot blue curve. It has almost the same profile with that of the right leg (solid blue) with an expected phase lag Force right leg Force left leg Average value Fig. 9 Force along z: gait sl=.3m, =.5s, =.2s According to the average force value of both of legs the z force distribution to the two legs is 58% to the back leg and 42% to the front leg. Accordingly, during the first double support phase is set to and during the second double support phase is equal to.following the same approach has been evaluated for the x:during the first double support phase and during the second support phase. B. Tuning The model developed is based on some assumptions (section III) which allow reducing the model complexity but at the same time they may affect the accuracy of the system. As mentioned above in order to tune the model,, and coefficients has been inserted in the model equation. In Fig. 1 the red line depicts the experimental motion of the COM, the blue line represents the reference trajectory of the COM and the output of the model before tuning (coefficients,, and, were set to one in this trial) before tuning before tuning Fig. 1 x and z motion of the COM before tuning. The reference trajectory sent to the robot was also used as an input reference of the model. Moreover the model receives the initial position and velocity at the beginning of the first double support phase. The experimental motion of the COM was derived from the joint angles measured by the joint encoder. The capability of the model to represent the behaviour of the robot was evaluated by the Mean Square Error (MSE) (24) where is the number of samples, and and are the experimental and the simulated trajectory point of sample. The MSE measured before the tuning of the parameters is approximately along x direction and along z direction. Those values have been computed over an interval of time corresponding to three consecutive steps (excluding swing phases). After this initial evaluation and in order to improve the model accuracy the following tuning procedure was performed in the model. Referring to Fig. 6 the whole system can be considered like two subsystems interconnected. and are outputs of and inputs of and similarly and are output of and input of. Initially, the two subsystems were decoupled. By feeding with experimental samples data as and input to the first subsystem and as and input to the second subsystem the coefficients and that minimize the MSE were adjusted. Similarly, and were determined. Following this and to identify the other parameters the interconnections between the two subsystems were activates in just one way (from the first system to the second). Taking and from experimental data and feeding them to and taking and from the first submodel to, the and were estimated. Finally and were adjusted by activating the interconnection from the second subsystem to the first. Table 1 reports all the values of the coefficients estimated with this procedure. TABLE I MODEL PARAMETERS Fig. 11 compares the model response after the tuning procedure (obtaining with the same reference trajectory of Fig. 1) with the experimental data recorded from the robot. With the tuning applied the MSE reduced to along X direction and along Z direction C. Validation Fig. 11 and real system comparison: Test 1 gait sl=.3m, =.5s, =.2s The model was also validated with other trajectories. Tests are performed by increasing the step length and changing the time duration of the walking phases. The MSE was not affected significantly when changing the walking gate. In Fig. 12 the blue line is the reference trajectory, the black is the true trajectory performed by the robot and in red the model result. As it is evident the accuracy was not affected significantly. The MSE in this test was along x direction and along z direction. 693

7 Z [m] X [m] Fig. 12 and real system comparison: Test 2 gait sl=.6m, =.6s, =.2s V. CONCLUSIONS AND FUTURE WORK The development of actuation systems with intrinsic passive compliance is the step towards the development of soft humanoid robot bodies which can inherently adapt to interaction uncertainties. Although the benefits gained with the incorporation of the physical elasticity are obvious the complexity of the system model increases significantly making inadequate the models developed in the past for equivalent stiff humanoids. This work proposed a reduced model for describing the motion behaviour of the compliant humanoid robot COMAN. The model is based in the inverted pendulum model augmented by the incorporation of the equivalent physical compliance due to the joints elasticity. In particular, the dynamics of the COM motion are described by set of dynamic equations which formulated using single mass model and the corresponding Cartesian stiffness and damping matrices at the COM. The derivation of the model was presented and its validation through experimental trials was demonstrated. It has been shown that this reduced model can describe with good efficacy the behaviour of the COM of the humanoid. Further developments will include the adaptation of the model to improve the tracking of the reference input trajectory and the development of control schemes for stabilization which will take into account the COM motion model to improve the regulation performance. A centralized model based controller can be potentially implemented using this representation of the robot behaviour with the computational effort reduced with respect to the complete dynamic model of the robot. Moreover in order to improve the precision of the model learning techniques could be adopted to compute the parameters of the model. ACKNOWLEDGMENT This work is supported by the European Commission FP7, AMARSI Project ICT REFERENCES [1] M. Hirose, Y. Haikawa, T. Takenaka, and K, Hirai, Development of Humanoid Robot ASIMO Proc. IEEE/RSJ IROS 21, Workshop2. [2] K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka, The Development of Honda Humanoid Robot, Proc. of IEEE ICRA 1998, pp [3] K. Akachi, K. Kaneko, N Kanehira, S. Ota, G. Miyamori, M. Hirata, S. Kajita and F. Kanehiro, Development of humanoid robot HRP- 3P, Proc. of IEEE-RAS Int. Conf. on Humanoid Robots, pp [4] Y. Ogura, H. Aikawa, K. Shimomura, A. Morishima, H. Lim, and A. Takanishi, Development of a New Humanoid Robot WABIAN-2, Proc. of IEEE ICRA 26, pp [5] N.G. Tsagarakis, G. Metta, G. Sandini, D. Vernon, R. Beira, F. Becchi, L. Righetti, J.S. Victor, A.J. Ijspeert, M.C. Carrozza and D.G. 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