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Single flexible joint impedance
During grasping tasks it is more advantageous to use a joint impedance controller than a joint position controller. Indeed, because the models of the objects are inaccurate, a position controller can lead to large interaction forces. Large forces can damage the fingers or the objects. Consequently, a force control loop is required to softly interact with the environment. Several control schemes have been developed that allow to moderate the forces, such as hybrid force control, admittance control or torque control. Based on the practical experience in manipulation of the DLR, a joint impedance controller is selected to provide the compliant behavior. Similar work is proposed in [68], however with experimental results on a very different system (an arm with comparatively large stiffness) as well as limited to the case of a constant stiffness.1 In this section, the backstepping design method is applied to a single joint driven by a single motor. Unlike the previous section, the behavior of an impedance controller is targeted. First, the coordinates of the joint model are transformed in order to apply the desired link side control law. The internal dynamics of the motor results in a tracking error of the desired link torque, therefore, the backstepping procedure is applied to ensure that the system is regulated to the desired state while remaining globally stable. In the third section, several simulations are presented in order to obtain a first selection of gains for the experiments. The practical implementation is presented in the last section. A test setup with a single flexible joint and adjustable parameters, such as link mass and joint stiffness, is designed and built for those specify tests.
Single flexible joint
impedance non linear stiffness In the previous sections, the spring stiffness was considered constant. However, in the Hand Arm System, nonlinear springs are used in order to offer the possibility to adjust the joint stiffness. Moreover, the explosion of the spring stiffness when reaching its elongation limits creates a natural protection for the end stops of joints. Preliminary experiments on a system with nonlinear springs with the backstepping controller designed for a linear spring showed that the controller is robust to the unmodeled nonlinearities. However, it is possible to include the nonlinear effects directly in the controller to ensure that the stability is achieved without the robustness properties. In this section the backstepping impedance controller for a single flexible joint is modified to include the nonlinear spring characteristic. First, the nonlinearity is introduced in the model. Then, the nonlinear effects are propagated in the controller. The nonlinear effects are only modifying the last backstepping stage. Unsurprisingly, one condition for the stabilit proof is that the spring stiffness is strictly positive. Finally, simulations and experiments are performed in order to verify the validity of the controller.
Simulations
In this section numerical simulation are performed to verify that the controller provides an impedance behavior and that it is stable (naturally limited to numerical experiments). For the simulation, a realistic spring characteristic is used and the derivatives are tabulated in a lookup table in order to stay close to the implementation case. Throughout this section, the term linear controller refers to the backstepping controller designed for the linear plant, the term nonlinear controller refers to the backstepping controller designed for the nonlinear plant. The difference between the linear and the nonlinear controllers is depicted in Figure 15.19. The improvement of the nonlinear controller is mainly noticeable in terms of settling time, although a larger overshot is observed. The nonlinear components of the controller have an effect only when the stiffness of the link is far from the nominal stiffness, therefore the difference between the trajectories is not noticeable during the free motion between t = 0.5s and t = 1.0s. Indeed, the inertia of the link is low w. r. t. to the joint stiffness thus the link deflection is minimal and the stiffness variation is negligible. A load applied to the link modifies noticeably the stiffness, this effect in depicted in Fig. 15.19 where the stiffness during both experiments is reported. Since a torque peak must be generated to begin the motion, one would expect a change of stiffness at the point denoted by A. However, this change of stiffness is negligible w. r. t.approaches, the cancellation of the nonlinear terms tends to generate very large control actions. Therefore, a comparison between the linear and the nonlinear controller, together with an torque input saturation, is reported in Fig. 15.21.
Experiments
Using the same setup as used in the previous section but replacing the linear springs by nonlinear ones, experiments are performed to verify that the controller behaves as expected with the physical plant. Similar to the linear case, the controller successfully moves the link to the desired position and provides an impedance behavior w. r. t. the externally applied load. The link side position and the joint torques trajectories are depicted in Figure 15.22. limited if the input saturation is taken into account. Although the stiffness of the joint is nonlinear, the stiffness of the link only changes significantly when an external load is applied. Simulations and experiments confirm that the controller performs satisfactorily and reveals to be very robust to disturbances. The implementation of this nonlinear backstepping controller requires a stiffness model and its derivatives. It is interesting to note that one condition for the use of the controller is that the stiffness and its first derivative are non zero.
Antagonistic joint
The previous sections have demonstrated that the backstepping method is able to provide a solid theoretical background as well as excellent practical results. However, the previous cases where limited to the case of a single joint driven by a single motor. As presented in the modeling part, the fingers of the Awiwi Hand are driven by an antagonistic arrangement of tendons thus, it is necessary to extend the backstepping controller to the case of an antagonistic controller. In this section, the backstepping design method is applied to a single joint driven by a pair of motors. First, the dynamics equations of the system are
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