Modeling

Kinematics, Dynamics, Control, and Simulation

this page last updated Oct 2003

	Derivations
	Simulation Software

	Open-Loop Simulation
	Open-Loop Animation

	Trenching Trajectory
	Trenching Animations

	Joint tracking simulation
	Model Validation

Derivations

The equations that describe the motion of the backhoe's links (swing, boom, stick, and bucket) are derived from the LaGrangian dynamic model. For a derivation of this model, as well as kinematic relationships, see the report on the backhoe Kinematics and Dynamics.

See the document Kinematics for the initial derivation of the kinematic transformations between cylinder lengths, joint angles, base frame and local frames, geometric Jacobian, reverse displacement, and a digging simulation.

The electrohydraulic proportional valve used on the robotic backhoe is modeled empirically from the results of Hardware-In-the-Loop testing.

Simulation Software

In order to capture all the significant dynamics of the backhoe during a digging operation, the mathematical models have been programmed using Simulink, a graphical programming environment well suited for control applications as well as dynamic nonlinear computations. The picture at the right is a snapshot of the top-level block diagram of this model. In Simulink, each block represents a function, which can be anything from a simple gain multiplier to a complex user-defined program hundreds of lines long. Variables are passed between functions in a variety of formats including scalars, vectors, matrices, or others, which are calculated and passed to the next function at each time step. More detail on the contents of each block are given in the fall 2003 presentation.

In the backhoe model, functions have been written in a modular format to compute all the relevant variables, such as oil flow through valves, joint accelerations, control signals, hose compliance, bucket-soil interaction, displacement transformations, cylinder pressures, joystick dynamics, haptic forces, etc. Since the system has four degrees of freedom, most of the lines passing variables between each function block are four-element vectors. The input to the model is the desired trajectory of the bucket in the mind of the operator; the output is, among other things, the bucket position and orientation vs. time. Once a simulation has been run, the time history of each dynamic variable is then available for analysis.

This model will be used as a design and analysis tool during the development of the Robotic Backhoe with Haptic Display.

Open-Loop Simulation

The open-loop plant in the simulation blocks shown above consist of the valves, hoses, cylinders, and backhoe links. Combining just these blocks and sending voltage pulse commands to each of the valves, the results of an open-loop simulation are shown below.

Open-loop valve commands

3D animation of joint trajectories (avi 1.6M)

ProE animation of simulation results (mpg 2.2M)

Trenching Trajectory

Using a subset of the blocks from the model above, a program was written to compute the trajectory of each joint as the backhoe moved through a typical trenching operation. Input to the program was the desired digging dimensions defined over a sequence of 13 steps: reach, depth, unload position, etc. The output from the program was the position and angle of each joint versus time, which is illustrated in the plots below.

2D view of joint trajectories

3D view of bucket and wrist trajectories

3D animation of joint trajectories (avi 1.3M)

Trenching Animations

Using the results from above, the joint angle histories were then read into the modeling program (Pro/ENGINEER) and used to define joint drivers to animate the solid model. The movie files below illustrate the solid model animation.

trench_des1 (mpg 1.4M)

trench_des2 (mpg 1.5M)

trench_des3 (mpg 1.7M)

These animations were generated using the desired trajectories. In the future, the solid model will be used to illustrate the output of the complete system simulation described above.

Joint Tracking Simulation

The plots below illustrate a trenching simulation. The plot on the left is the reference trajectory in the vertical x0-z0 plane, and the plot on the right illustrates the corresponding reference and model output in joint space. Although the plot on the right may seem trivial, considerable effort went into producing it. All told, the model includes 32 integrations per time step: three per valve, two per cylinder, two per link, and one integral control = 8 per degree of freedom x 4 degrees of freedom = 32 integrations per time step--i.e. a 32nd order model.

cartesian 2-D reference

joint angle tracking

Model Validation

The plots below compare the input/output data collected from the backhoe to the model prediction when the cylinders were given sinusoidal reference commands and an arbitrary digging trajectory, respectively. Click here to see video clips of the backhoe when the data was collected. Input is the voltage command sent to the valve and output is the measured cylinder lengths from the sensors. Distortion of the command signals is due to a combination of disturbances including unequal cap- and rod-side cylinder areas, vibrations, friction, and gravity. As can be seen, the model does a good job of simulating the backhoe's dynamics.

sinusoidal reference

trenching trajectory reference