This article proposes a control method for underactuated cartpole systems using semi-implicit cascaded proportional-derivative (PD) controller. The proposed controller is composed of two conventional PD controllers, which stabilizes the pole and the cart second-order dynamics respectively. The first PD controller is realized by transforming the pole dynamics into a virtual PD controller, with the coupling term exploited as the internal tracking target for the cart dynamics. Then, the second PD controller manipulates the cart dynamics to track that internal target. The solution to the internal tracking target relies on an equation set and features a semi-implicit process, which exploits the internal dynamics of the system. Besides, the design of second PD controller relies on the parameters of the first PD controller in a cascaded manner. A stability analysis approach based on Jacobian matrix is proposed and implemented for this fourth-order system. The proposed method is simple in design and intuitive to comprehend. The simulation results illustrate the superiority of proposed method compared with conventional double-loop PD controller in terms of convergence, with the theoretical conclusion of at least locally asymptotic stability.

The balance control problem of planar bipedal robots during disturbed standing is investigated. Virtual holonomic constraints which specify the angles of actuated joints as a function of the rotation angle between the sole of the stance foot and the ground are recalled. These constraints enable the robot to avoid turnover during external disturbances similar to how a human being would react. Moreover, for a disturbance beyond the balance conditions for single-leg support, the robot changes to a double support posture to avoid turnover by the virtual holonomic constraints. Further, for a special kind of balance: standing on toe which leads to underactuation, we also give the design conditions of the virtual holonomic constraints.

In this paper, a model-based exponential stabilization of a quadruped robot is studied in bounding motion. The dynamics of the five-link planar underactuated mechanical model of the quadruped robot with four actuated joints system is derived. It is shown that the dynamical equation of the proposed simplified model belongs to a class of second-order nonholonomic mechanical systems which cannot be stabilized by any smooth time-invariant state feedback. Utilizing a coordinate transformation based on the so-called normalized momentum, a robust backstepping control method is presented for the quadruped robot. Both theoretical analysis and numerical simulations show that the robust backstepping controller can stabilize the underactuated quadruped robot so that it could balance on its rear legs and track a desired trajectory. Despite the model parameter uncertainties, the robustness of the controller is maintained. The simulation results show the effectiveness of the proposed method.

This paper investigates the stable-running problem of a planar underactuated biped robot, which has two springy telescopic legs and one actuated joint in the hip. After modeling the robot as a hybrid system with multiple continuous state spaces, a natural passive limit cycle, which preserves the system energy at touchdown, is found using the method of Poincaré shooting. It is then checked that the passive limit cycle is not stable. To stabilize the passive limit cycle, an event-based feedback control law is proposed, and also to enlarge the basin of attraction, an additive passivity-based control term is introduced only in the stance phase. The validity of our control strategies is illustrated by a series of numerical simulations.

This paper presents a task-priority motion planning algorithm for underactuated robotic systems. The motion planning algorithm combines two features: the idea of the task-priority control of redundant manipulators and the endogenous configuration space approach. This combination results in the algorithm which solves the primary motion planning task simultaneously with one or more secondary tasks ordered in accordance with decreasing priorities. The performance of the task-priority motion planning algorithm has been illustrated with computer simulations of the motion planning problem for a container ship.

A Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.