Generic Model Implementation
The Compliant Joint Toolbox comprises linear models of both, the mechanical actuator subsystem and the electrical actuator subsystem as core components.
The most common electrical drive in torque controlled robotic actuators are brushless DC motors (BLDC), which can be operated such that the actual three phase motor dynamics are well described by a single phase approximation. The governing parameters are the electrical resistance and inductance. In torque controlled electrical robotic actuators the inductance is typically designed to be low. As a consequence, the electrical time constant becomes very small (≈ 10 −6 s) compared to the mechanical time constants (≈ 10 −3 s). Unless it is intended to specifically analyse the current control performance or its implications on higher level controllers, the electrical dynamics can be neglected with respect to the mechanical time constant. This substantially shortens simulation time. Hence, a static model is used in the built actuator models by default.
The Mechanical Subsystem: The mechanical subsystem is modelled as a chain of rotating masses interconnected via massless spring-damper elements as depicted in Fig. III-A2 The electrical drive rotor is an inertia I_m , which experiences a damping d_m with respect to the mechanical ground. The gearbox shows an inertia I_g and can be compliant with a linear stiffness k_g and internal material damping d_mg . Moreover, gear friction with respect to the mechanical ground is captured by d_g . The second elastic element is represented by massless torsional spring with linear stiffness k_b and internal material damping d_gl . Finally, the rotary inertia I_l models the load with frictional damping d_l . The motor, gearbox and load angles are denoted by q_m , q_g and q_l . The torques acting on the system are τ_m , τ_g and τ_l .
The linear equations of motion for this three-mass-system are straightforward to derive from first principles and can even be found in many textbooks on control or structural dynamics. The Compliant Joint Toolbox features several variants to this general model structure, such as the case of rigid gearbox, complete rigidity (single moving mass with friction) and locked actuator output configurations model. The following figure summarizes the model variants implemented in the Compliant Joint Toolbox.
The locked actuator output case emulates an infinitely high load inertia. This scenario is often used for torque controller design and analysis.
The Compliant Joint Toolbox implements the linear mechanical dynamics in state space form as illustrated above with the state vector x_q, system matrix A_q as well as input and output matrices B_q , C_q and direct feed-through matrix F_q. The input and output vectors of this model are denoted by u_q and y_q. The joint model has in total two inputs and generally seven outputs. The two inputs are the motor current along with a disturbance input that is either a load torque τ_l or load motion q_l . The first three elements of the output vector are the three angles q_m , q_g and q_l and the elements four to six are their derivatives. For convenience, the seventh output is the sensed joint torque that computes from the difference between the load angle q_l and gearbox angle q_g multiplied by the sensor stiffness parameter k_b.
The benefit of Compliant Joint Toolbox here is clearly that the user can simulate the dynamics and quickly switch between mechanical and electrical model structures or compare the actuator model structures against each other for identical parameter sets and within identical control schemes without manipulating equations and commenting/un-commenting or duplicating source code.
The additive nonlinear dynamics term g(x_q ) augments the linear state space model of the mechanical subsystem and modulates its input / output behaviour. This approach allows to capture a broad variety of practically relevant effects. From this perspective, the figure above illustrates this realization.
The most dominant nonlinear dynamics effect in torque controlled actuators is friction. The parameters d_m , d_g and d_l describe the symmetric linear viscous friction behaviour in the support and transmission mechanisms, which may also be observed to be asymmetric with respect to the sign of the velocity.
In addition to viscous friction, constant Coulomb friction is a nonlinear effect that dominates especially the lower speed regime of torque controlled actuators. While the asymmetry of viscous friction is often small, it tends to be significant with Coulomb friction.
Apart from friction, torque ripples perturb the actuator torque generation. Multiple sources contribute to the effect. Commutation ripple, mutual torque ripple, cogging torque ripple, current offset ripple, gearbox teeth meshing ripple, assembly eccentricity, encoder ripple. All ripple sources accumulate to a ripple torque τ_r that is periodic with the rotor angle q_m. The Compliant Joint Toolbox incorporates ripple through a Fourier series in the rotor angle q_m . This ripple model is linear in the amplitude parameters A_j and B_j and considers a number N_ω of spatial ripple frequencies ω_q. As all nonlinear dynamics terms result in an additional torque, they can be introduced into the models as an additional summand in the state equation as indicated in the figure above.
A prior use case of the Compliant Joint Toolbox is to simulate the nominal joint actuator behaviour to conceptually test and analyse controllers under ideal conditions. In the non-ideal case, the actual system input and output are each subject to additive noise. The communication interfaces with the hardware introduce delays in the commands.
