We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.

A short term goal in the forest industry is semi-automation of existing machines for the tasks of logging and harvesting. One way to assist drivers is to provide a set of predefined trajectories that can be used repeatedly in the process. In recent years much effort has been directed to the design of control strategies and task planning as part of this solution. However, commercialization of such automatic schemes requires the installation of various sensing devices, computers and most of all a redesign of the machine itself, which is currently undesired by manufacturers. Here we present an approach of implementing predefined trajectories in an open-loop fashion, which avoids the complexity of sensor and computer integration. The experimental results are carried out on a commercial hydraulic crane to demonstrate that this solution is feasible in practice.

We consider a 24-degrees-of-freedom monkey robot that is supposed to perform brachiation locomotion, i.e. swinging from one row of a horizontal ladder to the next one using the arms. The robot hand is constructed as a planar hook so that the contact point about which the robot swings is a passive hinge. We identify the 10 most relevant degrees of freedom for this underactuated mechanical system and formulate a tractable search: (a) introduce a family of coordination patterns to be enforced on the dynamics with respect to a path coordinate; (b) formulate geometric equality constraints that are necessary for periodic locomotion; (c) generate trajectories from integrable reduced dynamics associated with the passive hinge; (d) evaluate the energetic cost of transport. Moreover, we observe that a linear approximation of the reduced dynamics can be used for trajectory generation which allows us to incorporate the gradient of the cost function into the search algorithm.

We consider a 3-link planar walker with two legs and an upper body. An actuator is introduced between the legs, and the torso is kept upright by torsional springs. The model is a 3-DOF impulsive mechanical system, and the aim is to induce stable limit-cycle walking in level ground. To solve the problem, the ideas of the virtual holonomic constraints approach are explored, used and extended. The contribution is a novel systematic motion planning procedure for solving the problem of gait synthesis, which is challenging for non-feedback linearizable mechanical systems with two or more passive degrees of freedom. For a preplanned gait we compute an impulsive linear system that approximates dynamics transversal to the periodic solution. This linear system is used for the design of a stabilizing feedback controller. Results of numerical simulations are presented to illustrate the performance of the closed loop system.

Ball dribbling is a central element of basketball. One main challenge for realizing basketball robots is to stabilize periodic motions of the ball. This task is nontrivial due to the discrete-continuous nature of the corresponding dynamics. The ball can be only controlled during ball-manipulator contact and moves freely otherwise. We propose a manipulator equipped with a spring that gets compressed when the ball bounces against it. Hence, we can have continuous-time control over this underactuated Ball-Spring-Manipulator system until the spring releases its accumulated energy back to the ball. This paper illustrates a systematic way of planning such a modified dribbling motion, computing an analytical transverse linearization and achieving orbital stabilization.

A new approach for solving an optimal control problem of ball pitching with an underactuated human-like robot arm is proposed. The system dynamics is simplified to a planar two-link robot with actuation only at the shoulder joint and a passive spring at the elbow joint representing the stiffness of the arm. The objective is to accelerate the ball from an initial configuration at rest in such a way that the projection of its velocity along a certain elevation angle is maximal at a predefined release line. The suggested procedure makes use of a parameterization of the robot motion in terms of geometric relations among the generalized coordinates. We systematically formulate a necessary condition for an optimal motion resulting in a nonlinear differential equation that describes a synchronization of the joint angles. A suitable solution is found by numerically searching over a finite number of free initial conditions.

Ball dribbling is a central element of basketball. One main challenge for realizing basketball robots is to stabilize periodic motion of the ball. The task is nontrivial due to the discrete-continuous nature of the corresponding dynamics. This paper proposes to add an elastic element to the manipulator so the ball can be controlled in a continuous-time phase instead of an intermittent contact. Optimal catching and pushing trajectories are planned for the underactuated system based on the virtual holonomic constraints approach. First experimental studies are presented to evaluate the approach.

In the forest industry, trees are logged and harvested by human-operated hydraulic manipulators. Eventually, these tasks are expected to be automated with optimal performance. However, with todays technology the main problem is implementation. While prototypes may have rich sensing information, real cranes lack certain sensing devices, such as encoders for position sensing. Automating these machines requires unconventional solutions. In this paper, we consider the motion planning problem, which involves a redesign of optimal trajectories, so that open loop control strategies can be applied using feed-forward control signals whenever sensing information is not available.

We consider trajectory planning for an underactuated 3DOF helicopter, using the virtual holonomic constraint approach. First we choose constraint functions that describe the configuration variables along a desired motion in terms of some independent parametrization variable. This lets us describe the closed-loop system by some reduced order dynamics, the solution of which gives a feasible trajectory for the desired motion. By using the method of transverse linearization for controller design, we achieve exponential orbital stability to a desired trajectory. Numerical simulations confirm this property and show good convergence to a desired periodic motion when initialized from a resting state.

The steps for modeling and control of a hydraulic rotary actuator are discussed. Our aim is to present experimental results working with a particular sensing device for angular position as a complement to pressure sensing devices. We provide the steps in experimental system identification used for modeling the system dynamics. The cascade controller designed contains an inner loop for an accurate tracking of torque while stabilizing position reference trajectories. The performance of this design is experimentally verified.

The problem of swinging up inverted pendulums has often been solved by stabilization of a particular class of homoclinic structures present in the dynamics of the standard pendulum. In this article new arguments are suggested to show how different homoclinic curves can be preplanned for dynamics of the passive-link of the robot. This is done by reparameterizing the motion according to geometrical relations among the generalized coordinates. It is also shown that under certain conditions there exist periodic solutions surrounding such homoclinic orbits. These trajectories admit designing feedback controllers to ensure exponential orbital stabilization. The method is illustrated by simulations and supported by experimental studies.

We consider trajectory planning for kinematically redundant manipulators used on forestry machines. The analysis of recorded data from human operation reveals that the driver does not use the full potential of the machine due to the complexity of the manipulation task. We suggest an optimization procedure that takes advantage of the kinematic redundancy so that time-efficient joint and velocity profiles along the path can be obtained. Differential constraints imposed by the manipulator dynamics are accounted for by employing a phase-plane technique for admissible path timings. Velocity constraints of the individual joints are particularly restrictive in hydraulic manipulators. Our study aims for semi-autonomous schemes that can provide assistance to the operator for executing global motions.

A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.

A teleoperation system has been developed for a hydraulic crane, of the type used on a forwarder vehicle, which travels off-road and collects logs cut by a harvester. The system developed consists of a 3D virtual environment, which allows the user to input a desired position for the crane tip using either the mouse or a joystick. The desired position is then transmitted (via UDP/IP) to a local control system. The crane is a redundant manipulator, so movements of the individual links are calculated using a pseudoinverse method, and controlled using PIDs with friction compensation. Encoder data from the crane links are continuously sent back to the user side, and the crane’s movement is visualized in the virtual environment. The system has been tested on a real forwarder crane, experimental results and a video of the system’s performance are provided.