Hartmut Bremer

Modeling and Control of a Pneumatically Driven Stewart Platform

Electrically driven Stewart platforms are used in the field of machine tooling and robotics, where very accurate positions have to be reached associated with heavy loads. In this paper we present a pneumatically driven Stewart platform powered by fluidic air muscles. Due to the elasticity of the muscles and air as driving medium, the robot is predestined for applications where compliance plays a major role. Compliant behavior is necessary for direct contact with humans. Fitness is an area, where this contact is given and a fast movement is needed for the body workout. Another field of application are simulators for computer games or 6D cinemas. To realize the six degrees of freedom (x, y, z, α, β, y ) for the Tool Center Point (TCP) there are six fluidic muscles. Due to the fact that the muscles are only able to pull on the platform, there is a spring in the middle that applies a compressive force to the moving part of the robot. The spring is a non modified spiral spring which is commonly used for the suspension of a passenger car. As a result of the kinematical model (inverse kinematics, forward kinematics) the workspace is optimized. To dimension and test the dynamical behavior, a Matlab/Simulink model is derived. This is done by applying the Projection Equation, a synthetical method for obtaining the equations of motions for multi body systems. Based on the dynamical model we develop a control concept in a cascaded structure (pressure control, linearization, position control). A laboratory setup is used to validate the simulation model. Both, simulations as well as experimental results demonstrate the success of the proposed concept.

A persistent method for parameter identification of a seven-axes manipulator

This paper presents a persistent method for the identification problem of open-chained robotic systems. Based on the Projection Equation, a new, direct method to collect the dynamic and friction parameters in linear form is worked out. However, in this form, linear dependencies in the parameters occur and they are canceled out with the help of the QR algorithm. The obtained linear independent parameters are the base parameters of the system. To ensure a good excitation, the identification is improved by using optimized trajectories defined by Fourier-series, taking also physical constraints into account. The evaluation of the dynamic robot parameters is realized with a least squares error optimization. Furthermore, the result strongly depends on a special choice of weighting matrices for the error. Experimental results for a seven-axes robotic system (standard six-axes industrial manipulator mounted on a linear axis) are presented in detail. Additionally, the influence of temperature effects to base parameter changes is discussed.

Efficient Online Computation of Smooth Trajectories Along Geometric Paths for Robotic Manipulators

This paper presents a fast computation method for time-optimal robot state trajectories along specified geometric paths. A main feature of this new algorithm is that joint positions can be generated in realtime. Hence, not only joint velocities and accelerations limits but also constraints on joint jerks and motor torques can be considered. Jerk limits are essential to avoid vibrations due to (not-modeled) gear or structure flexibilities. For the limitation of motor torques a complete dynamic robot model including Coulomb and viscous friction is used. The underlying optimal control problem is found by projecting the problem onto the geometric path. The resulting state vector contains path position, speed and acce- leration while path jerk is used as input. From optimal control theory it follows that the path jerk has to be chosen at its boundaries, which can be computed for each state in each step. Continuous state progress is assured via so called test trajectories which are additionally computed in each step. As an example the algorithm is applied to a six-axis industrial robot moving along a straight line in Cartesian space.

Fast Moving Flexible Robot Dynamics

Abstract Flexible gear-link combinations are considered for robot subsystem computation. In order to allow arbitrary robot design, basic motion of considered subsystem has to remain unrestricted, leading to the use of nonholonomic velocities . Its treatment via analytical methods is tedious. Therefore, a special technique for the application of projection equations is recommended, yielding immediately the partial differential equations . A RITZ approach is used in the sequel. Special attention is thereby given to the determination of 2nd order displacement fields which are needed for correct (partial) linearization.

A bipedal walking pattern generator that considers multi-body dynamics by angular momentum estimation

Typically, gait pattern generation for bipedal robots utilizes a simplified model, neglecting the rate of change of the angular momentum. In this paper a linear parameter optimization problem for this simplified model, extended by an estimation of the angular momentum, is proposed to generate stable walking patterns that can be solved online and still consider the nonlinear effects of the multi-body model. The scheme was used successfully to generate trajectories for a full-size humanoid robot.

Passivity Based Backstepping Control of an Elastic Robot

In industry plants, a trade off between the requirements of speed and cost efficiency often results in lightweight constructions. However, this introduces elastic deflections causing vibrations and a loss in tracking precision. Hence, investigations in suitable models and control laws are necessary. By using the Projection Equation with a Ritz expansion, a set of nonlinear ordinary differential equations which describes the motion of the system is developed. The utilized control law is a combination of a feedforward and a feedback scheme. The latter is based on backstepping methods, with respect to passivity ports. Finally experimental results are shown to verify the proposed control strategy.

Bipedal balancing control based on the centroidal momentum pivot and the best COM-CMP regulator

For stable walking of bipedal robots it is necessary to stabilize the unactuated degrees of freedom of the robot. Typically this is done by reducing the non-linear multi-body dynamics to a simple approximation and then controlling the linear momentum of the system. In this paper a feedback controller is proposed that also controls the angular momentum in a feedback loop while considering the full multi-body dynamics to extend the set of balanced states.

Online walking gait generation with predefined variable height of the center of mass

For biped robots one main issue is the generation of stable trajectories for the center of mass (CoM). Several different approaches based on the zero moment point (ZMP) scheme have been presented in the past. Due to the complex dynamic structure of bipedal robots, most of the considered algorithms use a simplified time invariant linear model to approximate the dynamics of the system. This model is extended to a time variant one and then used to generate stable CoM trajectories with variable predefined CoM height. This allows to generate trajectories online for walking underneath obstacles with more accuracy. It is shown that using this extended scheme it is possible to overcome some kinematic limits as joint speed in the knee or the maximum step length for common walking.

Dynamical Modeling of Flexible Linear Robots

The production sector aims towards increasing capacity and energy efficiency. A possibility to achieve this is the usage of manipulators built of lightweight structures in order to raise the maximum velocity, acceleration and payload. This leads to an elastic robot that tends to vibrate. The main focus of this paper is the modeling of a fast moving, elastic robot with three linear axes. These axes are connected by four flexible links driven by synchronous motors with elastic gear racks and bearing elasticities. The Projection Equation in subsystem form is used to calculate the dynamic model. This equation in combination with a Ritz approximation for the flexible links, which are modeled as Rayleigh beams, leads to a set of highly nonlinear ordinary differential equations. The usage of the Projection Equation in subsystem form simplifies the modeling of this system and offers the possibility of a fast numerical integration supported by the Maple ® packages SimCode 2, SimSubs and SimRecursive . The subsystems of the elastic bodies are assembled by the kinematical chain. This leads to the possibility to evaluate the minimal accelerations of the system by a recursive scheme with O(n) efficiency. These results for the endpoint (position and acceleration) of the complete elastically modeled robot are compared to experimental measurements.Copyright

A Penalty Shooting Walking Machine

In this paper, we consider the dynamical behavior and the control of a two legged Walking Machine. Special investigation is done in the evaluation of the equations of motion. Thus, for obtaining the accelerations of such a tree structured multi-body system, an O(n) algorithm is presented, enhancing the numerical stability. To fulfill the aim of modeling a gaitcycle, contact is opened and closed periodically. Therefore the O(n) algorithm is enhanced for this case. A controller is shown, that balances the robot when doing such highly dynamically processes like shooting a soccer penalty. Simulations and experimental results are presented.