38 4.3 Plot of the torque in the knee vs. final element, for the foot i. FIGURE 1.1: The lightly built, streamlined, agile body of the cheetah makes it an efficient sprinter.
Background
Modelling and simulation
For simple maneuvers, selecting control inputs can be as simple as fully activating a particular muscle. Direct single capture (and its extension, multiple direct capture) is a tool that uses forward integration to find control inputs programmatically.
Trajectory optimisation
Fortunately, the results obtained by trajectory optimization can be compared and combined with the results of other independent approaches to achieve certainty in the conclusions. While the setup time of an optimization problem is not easily negligible, the time to convergence is generally dominated by the product of the iteration time and the number of iterations in the optimizer used to solve the problem [38].
Problem statement
Others have made the problem easier for the optimizer by, for example, neglecting parts of the dynamics [35] (at the cost of solution accuracy), assuming that the feet do not slip, and prescribing (fixing) the order and timing of foot contact. It is worth noting that an aspect of trajectory optimization that has received little attention in terms of performance improvement is the kinematic formulation of the model itself.
Project objectives
Project scope
Project outline
Project outcomes
Research publications
- The manipulator equation
- Modelling drag
- Modelling input torques
- Modelling springs and dampers
The reverse operation (conversion from inertial to body frame) can be achieved by multiplying by the transpose of the rotation matrix, . Is the reference area, often defined as the orthographic projection of the object in a plane perpendicular to the velocity.
Trajectory optimisation
- Modelling contact
- Collocation
- Variable bounds
- Initialization and task-specific constraints
- Solving
- Metrics used
Next, the values at each element are used to seed the values at all collocation points of the corresponding element for the subsequent higher-accuracy Radau stage. The first metric is time, calculated as the sum of the length of all finite elements. This also has many variants - one approach is to minimize the peak of the total input power at each finite element, written as Jpeak =Pmax, where.
Software libraries
Method 20 distribution is expected to have a smaller variance since larger torque values contribute significantly more to the cost function. The plot compares the number of operations in the symbolic equations of motion (EOM) with the number of terms in. In the remainder of this chapter, we compare the performance of trajectory optimization problems formulated using relative versus absolute orientation coordinates.
Multi-body dynamics
Equations of motion
This causes the equations of motion and contact equations to have many more non-zero partial derivatives, resulting in poorer performance of the solver. As an illustration of how these improvements carry over to the equations of motion, Figure 3.2 compares the sparsity of each of the Coriolis terms (C ˙q ) for a 4-term plane pendulum. Absolute Angular Coordinates 24 Figure 3.3 compares the sparsity patterns for contact Jacobians and Hessian foot heights for the 3D monopod model described later in this chapter.
Joint constraints
The additional angular coordinates required in an absolute formulation are noted in gray and the constraint moment variables are. ˆxi·ˆyj =0 ˆxi·ˆzj =0 (3.2) They are implemented using the constraint momentsτcφandτcψ that act on the limited degrees of freedom. Depending on the NLP solution algorithm used, this change from variable bounds to constraint equations can affect the way these constraints are handled.
Experiments
Planar n-link pendulum swing-up
It also potentially adds more constraints, as range-of-motion constraints that can be set with variable bounds in a relative formulation now require equations to specify.
Planar monoped hopper
As with the pendulum, the absolute angle model used a smaller fraction of the solve time for the NLP feature estimates: it used an average of 20%, while the relative angle model used 46%. As in the previous tests, the 65% solution time spent evaluating the NLP function for the absolute angle model was significantly less than the 83% time spent for the relative angle model. Each leg of a quadruped contains the same degrees of freedom as a leg of a unipod.
Summary
The final pose on the right att = T is constrained to be the same as the initial pose att=0, but rotated by. Next, the rotation: the position and velocity data of a point sampled from the periodic corridor were used to establish the initial and final points, with the final position corresponding to a 60 degree swing rotation from the initial orientation. The final (x, y) position was unspecified since the amount of space required to perform the turn was not known.
Discussion
- Build time
- Maximal coordinates
- Accuracy
- Limitations of the study
The periodic walk for the absolute-reference quadruped reliably converged in less than 30 minutes, while the fast turn converged in one to three hours (depending on the initial seed). Without simplification, the 3D monopad can be built in 140 seconds for the absolute angle model or 155 seconds for the relative angle model. For the quadruped, the relative angle model could not be fully simplified within 12 hours, despite simplifying the equations in parallel on a 16-core computer.
Conclusion
These limitations can be removed, at the expense of accuracy for longer time horizon maneuvers where incremental errors can become significant. Indeed, it is possible that the difference in performance between the two configurations may not be as noticeable to a different solver. Although there were differences in the overall performance, the relative performance between the configurations did not change significantly.
Parameters
Pendulum
Absolute Angle Coordinates 32 whether the maximum formulation is beneficial in other cases, and whether there are modifications that can be made to improve its performance. There is no positional drive due to explicit enforcement of the system's geometry at the positional level.
Monopeds
Quadruped
Parallel to the contributions from chapters 3, 5 and 6, a software library was developed for trajectory optimization of legged animals and robots. This chapter describes the features, code format, and general function of the library, called physical_education. When modeling a system, the library uses the modeling change described in Chapter 3 to simplify system dynamics.
Code example
The software library for physical_education 38 Once the problem is solved or the time limit is reached, we need to check the final penalty value of the robot. FIGURE 4.2: Graph of the state of the body versus the finite element, for the example hopper with one foot. FIGURE 4.3: Graph of the torque input at the knee versus the finite element, for the foot in the monoped hopper example.
General code layout
The physical_education software library 40 defaults every finite element, but one can specify that interpolation can be used to get a constant time step by entering a value fordt. For documents like this, keyframes of the animation can be shown instead of a PDF.
Features and details of operation
- Automatic derivation of dynamics
- Nodes
- Plotting and animation
- Type hints
- General code
- Dummy and slack variables
Physical_education Software Library 42 These are used to derive the kinetic and potential energy of the system (EkandEp respectively) before calculating the M,CandG terms of the manipulator equation. TheNodeinterface ensures that each new component (spring, drag, and so on) can be implemented in its own file without necessarily changing the library's source code. This can result in significant performance gains due to improved numerical conditioning of the problem.
Discussion
- The drag model
- Modelling the tail
- Modelling the spine
- Modelling the legs
- Significant differences from real cheetah
There are two degrees of freedom and two input moments each at the base and center of the tail. Beginning with the construction of the model: the assumed equal distribution of mass within a given rigid body is known to be incorrect. FIGURE 5.6: Image of a cheetah with a partially transparent version of the model superimposed, intended to highlight similarities and differences.
Parameter identification
- Dimensions and masses
- Tail parameters and drag coefficients
- Contact parameters
- Actuator angle limits
- Power limits
- Actuator torque limits
It was added as a simple and somewhat reliable metric relating power, mainly as a bootstrapping method to estimate the actual torque limits of the individual actuators in the model. Set all other limits and parameters of the model as described in the previous chapter. Finally, increase the estimated limits by 20% to account for the fact that the cheetah is not necessarily at its peak in the dataset.
Conclusion
- Steady-state gallop
- Periodic velocity change gallop
- Constant-rate turn
- Turn initiation
- Discussion
The variables for the first half of the movement were initialized with the data from the canter used to determine the starting point. Similarly, body height was constrained to remain within 0.2≤ z ≤ 0.8, otherwise the model would. While the problem could be "fixed" (restrictions met and the contact penalty reduced to zero), the results were sub-optimal.
Measuring the effect of the tail
Tail activity
To begin, the activity A of the tail was quantified by finding the average of the absolute relative angular velocities in degrees per second, between the body and first link of the tail, and between the two links of the tail. 6.1) The metric is only used to compare activity between the full model and the variant without dragging its tail, as the value is not useful on its own. TABLE 6.1: Comparison of tail activity between cheetah models with and without tail drag, performing a periodic gallop. TABLE 6.2: Comparison of tail activity between cheetah models with and without tail drag, performing a constant-rate turn.
Energy usage
Discussion
A massless tail with drag could be tested to further compare the weight in the tail with the drag on the tail. This establishes progress towards answering the research question: can the role of the cheetah's tail be better understood via trajectory optimization applied to a large, complex model. Using this model, incremental progress was made toward understanding the cheetah's maneuverability at high speeds, including an indication of the role of the cheetah's tail during high-speed turns and other maneuvers involving large accelerations .
Future work
Optimal control to generate whole-body motion using center-of-mass dynamics for predefined multi-contact configurations. Small Change, Big Gains: Effect of Orientation Formulation on Solution Time for Multibody Path Optimization”. Combining the Advantages of Feature Approximation and Path Optimization.” In: Robotics: Science and Systems.
Average solve time and cost of all experiments
Average build time for all experiments
Parameters for each link in the 3D quadruped model
Table of constants for the drag model
Parameters for each link, modelled as a cylinder, in the cheetah model. 52
Comparison of tail activity between cheetahs models with and with-
Comparison of tail activity between cheetahs models with and with-
Energy usage for three cheetah models, performing a periodic gallop
