Modelling an industrial manipulator a case study
Introduction
This paper examines the modelling and validation of an open-chain rigid-body industrial manipulator (Fig. 1). In practice, this modelling step may occupy greater than 80–90% of the effort required in control systems analysis and design [1], [2]. Consequently, the significance of understanding the dynamic structure of the plant is found to be particularly important, and the framework is quite detailed. The manipulator model described in this paper was devised to enable an optimal minimum time path to be controlled.
In Section 3, the mechanical equations of motion of the robot device are derived, under the assumption (taken from the physical robot available for testing) of n rigid links joined together serially by n revolute joints. The decision to opt for the Lagrangian formulation of mechanics is also discussed, whilst the actuator characteristics of the DC servomotor driving each robot joint are described in Section 4. Although the dynamics of the joint actuators are, very often, overlooked in the manipulator controller design, the inability to specify joint torques in the typical servomotors of commercial arms would deem most advanced control strategies unsuitable, since almost all of them are invariably based on the capability to control joint torques. Hence, modelling the actuators is not only aimed at accurately simulating the most significant motor and driver dynamics, but to allow driving the actuators in torque mode, as specified by the controller design.
The robot is modelled using `TELEGRIP' software on a UNIX machine but the procedures described here could be implemented on a PC using `WORKSPACE' using a little more effort and would be completely scaleable.
Section snippets
System modelling
There are many conceptually different ways in which the components of the mechanical manipulator that need to be modelled, namely the motor characteristics, the kinematic parameters and the inertial (and may be load) parameters, can be obtained.
Both kinematic and inertial characteristics can be determined from design blueprints. Because machining has limited precision and assembly may be imperfect, there will always be small variations in kinematic parameters – link lengths that are a little
Mechanical modelling
There are a number of procedures for generating the dynamic equations of motion for a robot arm, i.e., the equations, which relate joint forces and torque τ(t) to positions θ(t), velocities and accelerations , in terms of the specified kinematic and inertial parameters of the links. At present, a number of ways have been proposed for this purpose, such as:
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Lagrange–Euler (L–E) method [5].
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Newton–Euler (N–E) method [6].
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Recursive Lagrangian method [7].
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Kane's method [8].
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Appel's method [8].
Electro-mechanical modelling
The previous section has described how the motor torques required to drive the manipulator arm can be obtained from known physical laws such as the Lagrangian formulation of mechanics. Eq. (15) computes the mechanical torque set point that serves as the input to the robot arm actuators. However, the inability of commercial robots to control joint torques is a well-known problem [1], [18]. Commercial motor servos are typically position controllers, to which one can only send positional
Dynamic simulation
The dynamic models developed in this work were employed not only in the design of the optimal controller, described in Miro [16], but also to simulate the behaviour of the robot manipulator for a commercial application to develop a radiochemical dispenser. Simulation models have traditionally been approached by textual-based computer simulation languages, both discrete (GPSS, SIMULA, etc.) and continuous (ACSL, CSMP, etc.) some of which provide, at most, the capability to plot some simulation
Discussion
Derivation of the dynamic model of the CRS A251 industrial new controller has been attempted using a validation technique to match parameter values to give adequate response. This was not limited to the body structure dynamics, which experimental results showed appropriately approximated by Lagrangian rigid body mechanics, but also to the often ignored motor and driver dynamics, which were reverse engineered to be able to drive the actuators in torque mode. To decide whether the model and the
Conclusions
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LE modelling of a CRS A251 5 axis robot gives measured agreement with test data for polynomial trajectories to within 5% for the first two joints but within 14% for the third joint.
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Validation using correlation coefficients for the motor-torque gives accuracy's of 1% for the first two axes and to within 8% for the third axis.
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Unmodelled vibrations are shown to yield natural frequencies greater than 10 Hz.
Acknowledgements
This research program was partly sponsored by British Nuclear Fuels Plc.
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