Finding Forces from Motions

About Inverse Dynamics

The SimMechanics™ Kinematics and Inverse Dynamics modes enable you to find all the forces on a closed-loop machine or an open machine, respectively, given a model that completely specifies the system's motions. (See Choosing an Analysis Mode)

You can use the Forward Dynamics mode to analyze inverse dynamics, but these two alternative modes are more efficient: unlike Forward Dynamics mode, they do not need to compute the positions, velocities, and accelerations of the model's components, because the model specifies them. Consequently, Kinematics and Inverse Dynamics modes take less time than Forward Dynamics to compute the forces on a system. The time saving depends on the size and complexity of the system being simulated.

The following sections show how to use the Inverse Dynamics and Kinematics modes to find the forces on the joints of an open- and closed-topology system, respectively.

Building Kinematic Models

To use these modes, you must first build a kinematic model of the system, one that specifies completely the positions, velocities, and accelerations of the system's bodies. You create a kinematic model by interconnecting blocks representing the bodies and joints of the system and then connecting actuators to the joints to specify the motions of the bodies.

Actuating Independent Degrees of Freedom

A model does not have to actuate every joint to specify completely the motions of a system. In fact, the model need actuate only as many joints as there are independent degrees of freedom in the system. (See Counting Model Degrees of Freedom.) For example, a model of a four bar mechanism need actuate only one of the mechanism's joints, because a four bar mechanism has only one degree of freedom. To avoid overconstraining the model's solution, the number of actuated joints should not exceed the number of independent degrees of freedom.

    Warning   Attempting to simulate an overconstrained model causes Simulink® to stop the simulation with an error.

Inverse Dynamics Mode with a Double Pendulum

Consider a double pendulum consisting of two thin rods each 1 meter long and weighing 1 kilogram. The upper rod is initially rotated 15 degrees from the perpendicular.

Suppose that you want the pendulum to follow a certain trajectory. How much torque is required to make the pendulum follow this prescribed motion? Solving this problem entails building a kinematic model of the moving pendulum.

  • The model must represent the geometry of the double pendulum and specify its motion trajectory throughout the simulation.

  • The model must also measure the computed torque on each joint, the torque necessary to reproduce the specified motion.

    Except in simple cases, you can find these computed torques only as approximate functions of time.

The kinematic model can take different approaches to specifying the initial state of the pendulum.

  • One approach uses Body block parameters to specify the initial states.

  • Another approach uses Actuator block signals.

Using Body Blocks to Specify Initial Conditions

Open the model mech_dpend_invdyn1mech_dpend_invdyn1. It illustrates the Body block approach to modeling initial states.

This model represents the pendulum by two Body blocks and two Revolute Joint blocks.

  • The CS1 axis of the upper body (B1) of the pendulum is rotated 15 degrees from the perpendicular (see annotation for block B1).

  • The coordinate systems for the lower block (B2) are aligned with CS1 of the upper block. The CS1 of B2 is rotated -15 degrees relative to CS1 of B1, i.e., it is perpendicular to the World coordinate system.

Using Actuator Blocks to Specify the Initial States

Open the model mech_dpend_invdyn2mech_dpend_invdyn2. It shows the use of Joint Actuator blocks to specify the initial kinematic state. Using actuators to specify the displacement slightly simplifies the configuration of the Body blocks.

Specifying the Motion and Measuring the Computed Torques

In either model, the Joint Actuator blocks connected to the Joint blocks specify that the upper and lower joints accelerate at two distinct rates, 2 and -1 degrees/second2, respectively. Sensor blocks connected to To Workspace blocks measure the computed torques on the upper and lower joints as MATLAB® workspace variables torque_upper and torque_lower, respectively. These vectors capture the upper and lower computed torques at each major time step. You must simulate either model in Inverse Dynamics mode to compute the joint torques required to maintain the pendulum in its motion.

Using the Computed Torques in Forward Dynamics

Once you know the computed torques as functions of time, you can verify that these are the correct answers by creating a version of the model that applies the computed torques to the joints and simulating that model in Forward Dynamics mode.

Open the model mech_dpend_actmech_dpend_act. It illustrates a forward dynamics version of the kinematic model that uses the joint actuators to specify the initial angular displacement of the pendulum bodies.

This model uses Initial Condition blocks to specify the initial 15 degree displacement of the upper body from the vertical in the world coordinate system and the corresponding initial -15 degree displacement of the lower body from the vertical in the coordinate system of the upper body. The negative displacement of the lower body is equivalent to positioning it as vertical in the world coordinate system.

From a MAT-file, the model loads the upper and lower torques, torque_lower_fcn and torque_upper_fcn, as two matrices representing discrete functions of time. Simulating this model in Forward Dynamics mode results in the following display on the upper joint scope.

If the computed torques were known exactly as continuous functions of time in the two inverse dynamics models, this plot would exactly match the upper joint motion in the original models. But the torques are measured only in a discrete approximation, and mech_dpend_act does not exactly reproduce the original motion.

Making More Accurate Torque Measurements

You can achieve better approximations by adjusting Simulink to report sensor outputs in the original models with finer time steps. Refer to the Simulink documentation for more about exporting simulation data and refining output.

Kinematics Mode with a Four Bar Machine

Consider the four bar system illustrated by the tutorial titled Model and Simulate a Closed-Loop Machine. The model is mech_four_barmech_four_bar.

Suppose that you want to keep this system from collapsing under its own weight. Because the four bar has only one degree of freedom, applying a counterclockwise torque to the joint labeled Revolute1 would accomplish this objective. But how much torque is sufficient?

To answer this question, you must build a kinematic model of the stationary four bar system, starting with the tutorial model. The kinematic model must specify how the system moves over time. In this case, the four bar remains stationary. You can use a Joint Actuator to implement this requirement.

Transforming Forward into Inverse Dynamics

Open the model mech_four_bar_kinmech_four_bar_kin, derived from mech_four_bar.

  • The model uses a Joint Actuator block driven by a Constant block to specify the motion on the Revolute1 joint. The Constant block outputs a three-element vector that specifies the angular position, velocity, and acceleration, respectively, of the joint as 0.

  • The model uses a Joint Sensor block connected to a Scope block to display the resulting torque on the joint and a To Workspace block to save the torque signal to the MATLAB workspace.

Finding and Checking the Needed Torque

Now obtain and verify the inverse dynamics solution to the question.

  1. Run this model in Kinematics mode. The output reveals that the torque on the Revolute1 joint is 27.9 newton-meters, to the precision of the assembly tolerances specified in the Machine Environment block.

  2. To verify that the computed torque is, indeed, the torque required to keep the system stationary, create a forward-dynamics model that applies the computed torque to the Revolute1 joint. Open such a model contained in mech_four_bar_statmech_four_bar_stat.

  3. Run the model in Forward Dynamics mode, with the Revolute1 Angle Scope open.

    The Scope display reveals that the machine does, indeed, remain stationary, although only for about 0.4 seconds. The derived computed force is not exact, and the model is begins nonlinear oscillations after this period.

    Tip   You can reduce the inaccuracy of the derived computed force by rerunning the mech_four_bar_kin model with more restricted solver, assembly, and constraint tolerances. For the highest accuracy (at greater computational cost), consider shifting to the machine precision constraint solver. See Configuring Methods of Solution.

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