Jaume Franch

Velocity and position control of a wheeled inverted pendulum by partial feedback linearization

In this paper, the dynamic model of a wheeled inverted pendulum (e.g., Segway, Quasimoro, and Joe) is analyzed from a controllability and feedback linearizability point of view. First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations. This model is compared with the previous models derived for similar systems. The strong accessibility condition is checked and the maximum relative degree of the system is found. Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated. The resulting equations are then used to design two novel controllers. The first one is a two-level velocity controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch (pendulum angle from the vertical) within a specified range. The second controller is also a two-level controller which stabilizes the vehicles position to the desired point, while again keeping the pitch bounded between specified limits. Simulation results are provided to show the efficacy of the controllers using realistic data.

Velocity control of a wheeled inverted pendulum by partial feedback linearization

In this paper, the dynamic model of a wheeled inverted pendulum (e.g. Segway (2003), Quasimoro (Salerno and Angeles, 2003), Joe (Grasser et al., 2002)) is analyzed from a controllability and feedback linearizability point of view. First, a dynamic model of this underactuated system is derived with respect to the wheel motor torques as inputs while taking the nonholonomic no-slip constraints into considerations. This model is compared with the previous models derived for similar systems. The strong accessibility condition is checked and the maximum relative degree of the system is found. Based on this result, a partial feedback linearization of the system is obtained and the internal dynamics equations are isolated. The resulting equations are then used to design a two-level controller for tracking vehicle orientation and heading speed set-points, while controlling the vehicle pitch within a specified range. Simulation results are provided to show the efficacy of the controller.

Design of differentially flat planar space robots: a step forward in their planning and control

The motion of free-floating space robots is characterized by nonholonomic, i.e., non-integrable rate constraint equations. These constraints originate from principles of conservation of linear and angular momentum. It is well known that these rate constraints can also be written as input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. However, under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally in-expensive way to plan trajectories for the dynamic system between two configurations as well as develop feedback controllers. Nonholonomic rate constraints for free-floating planar open-chain robots are systematically studied to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, the property of flatness is used for trajectory planning and feedback control under perturbations in the initial state.

Differential Flatness of a Class of

A fully actuated system can execute any joint trajectory. However, if a system is under-actuated, not all joint trajectories are attainable. The authors have actively pursued novel designs of under-actuated robotic arms which are both controllable and feedback linearizable. These robots can perform point-to-point motions in the state space, but potentially can be designed to work with fewer actuators, hence with lower cost. With this same spirit, the technical note investigates the property of differential flatness for a class of planar under-actuated open-chain robots having a specific inertia distribution, but driven by only one or two actuators. This technical note addresses the following theoretical question: what placement of one or two actuators will make an n-DOF planar robot differentially flat if it is designed so that its center of mass always lies at joint 2?

n

Deals with the problem of linearization by prolongations of two-input driftless systems. For general two-input systems, the number of computations needed to check if a system is linearizable by prolongations is quite large. However, for driftless systems, the conditions presented in the paper require very few computations. The methodology is illustrated for some engineering systems which fulfill these conditions, e.g., a unicycle, a planar robot, and a hopping robot.

-DOF Planar Manipulators Driven by 1 or 2 Actuators

This paper gives a bound on the number of integrators needed to linearize a control system with an arbitrary number of inputs. Although some work has been done in this direction, our bound improves the existing results for systems with four or more inputs. The bound for two input systems is the same as the one that appeared in the literature. The bound for three input systems has been improved further in a previous paper.

Linearization by prolongations of two-input driftless systems

This paper demonstrates that for certain choices of mass distribution and addition of springs, an underactuated two-degree-of-freedom (2-DOF) \bm PRRRP system is static feedback linearizable, i.e., differentially flat as well. This paper is original and provides a ground breaking study in underactuated dynamical systems.

Linearization by prolongations : New bounds on the number of integrators

The motion of free-floating space robots is characterized by non-holonomic constraints, i.e., non-integrable rate constraint equations. These constraints originate from the principles of conservation of angular momentum. It is well known that these rate constraints can also be written to form an input-affine drift-less control systems. Trajectory planning of these systems is extremely challenging and computation intensive since the motion must satisfy differential constraints. Under certain conditions, these drift-less control systems can be shown to be differentially flat. The property of flatness allows a computationally inexpensive way to plan trajectories for a dynamic system between two configurations as well as develop feedback controllers. The key contribution of this paper is to systematically study the non-holonomic rate constraints for free-floating planar open-chain robots and determine the design conditions under which the system exhibits differential flatness. A design is then proposed that can exploit the property effectively for trajectory planning and feedback control.

Differentially Flat Design of a Closed-Chain Planar Underactuated

It is well known that a controllable nonlinear system will retain its controllabality when new actuator inputs are added to it. In this article, we ask the question if a system, linearisable by static or dynamic feedback, will retain this property when new actuator inputs are added to it. Alternatively, a system may be linearisable after removing one or more inputs from it. This question is important in the design of robotic systems from the perspective of trajectory planning and control, specially if they are under-actuated. The goals of this article are as follows: (i) using counter examples, we first show that feedback linearisability may not be preserved when new inputs are added to a robotic system, (ii) sufficient conditions are determined when a system will retain this property under the addition of new inputs. The theory is illustrated through some examples from the robotics field.

\hbox{2}

Gives a bound on the number of integrators needed to linearize a control system with an arbitrary number of inputs. Although some work has been done in this direction in Sluis and Tilbury (1996), our bound improves the existing results for systems with four or more inputs. The bound for two input systems is the same as the one that appeared in the above mentioned paper.