1.2.1 Review LTI Mechanical Dynamical Systems

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Из курса от партнера University of Pennsylvania

Robotics: Mobility

430 оценки

University of Pennsylvania

Robotics: Mobility

430 оценки

Курс 3 из 6 — Specialization Robotics

How can robots use their motors and sensors to move around in an unstructured environment? You will understand how to design robot bodies and behaviors that recruit limbs and more general appendages to apply physical forces that confer reliable mobility in a complex and dynamic world. We develop an approach to composing simple dynamical abstractions that partially automate the generation of complicated sensorimotor programs. Specific topics that will be covered include: mobility in animals and robots, kinematics and dynamics of legged machines, and design of dynamical behavior via energy landscapes.

Из урока

Introduction: Motivation and Background

We start with a general consideration of animals, the exemplar of mobility in nature. This leads us to adopt the stance of bioinspiration rather than biomimicry, i.e., extracting principles rather than appearances and applying them systematically to our machines. A little more thinking about typical animal mobility leads us to focus on appendages – limbs and tails – as sources of motion. The second portion of the week offers a bit of background on the physical and mathematical foundations of limbed robotic mobility. We start with a linear spring-mass-damper system and consider the second order ordinary differential equation that describes it as a first order dynamical system. We then treat the simple pendulum – the simplest revolute kinematic limb – in the same manner just to give a taste for the nature of nonlinear dynamics that inevitably arise in robotics. We’ll finish with a treatment of stability and energy basins. Link to bibliography: https://www.coursera.org/learn/robotics-mobility/resources/pqYOc

1.2.1 Review LTI Mechanical Dynamical Systems26:20

1.2.2 Introduce Nonlinear Mechanical Dynamical Systems: the dissipative pendulum in gravity22:51

1.2.3 Linearization & Normal Forms11:44

Познакомьтесь с преподавателями

Daniel E. Koditschek
Professor of Electrical and Systems Engineering
School of Engineering and Applied Science

In this section, we're going to review the study of kinematics,

which is related to the motion of bodies.

This is distinct from dynamics,

because we don't consider the forces that cause the accelerations of these bodies.

First, we define the phrase degrees of freedom.

These are the number of parameters that fully define the system in its space.

A rigid body and ambient space has six degrees of freedom.

Typically, they are x, y, and z, which are three positions, and roll,

pitch, and yaw, which are three angles.

In this figure, we see a plane who's degrees of freedom

have been highlighted in the typical convention.

In robotics, we're interested in both active degrees of freedom,

which are explicitly controlled by actuators, as well as the passive degrees

of freedom that are governed by the system's dynamics.

Most robots are highly underactuated.

This means that the majority of the degrees of freedom are passive and

not controlled by the system's actuators.

In this video, we see the pendulum robot.

The body of the robot has only two degrees of freedom,

as it is constrained on a boom.

The leg also has two degrees of freedom.

One of which, the rotation, is active as it is controlled by the motor.

The other is the extension of the leg,

which is simply governed by the attached spring.

Joints are an example of a holonomic constraint and

reduce the number of degrees of freedom of a system.

In this video,

we see a leg from the minitaur robot demonstrating multiple revolute joints.

In this video, we see a similar leg jumping on a linear rail.

All of the pin joints in the minitaur leg and

the linear rail constrain the attached links to only one degree of freedom.

Another example of a joint that has one degree of freedom is a screw

even though the translation and rotation are coupled.

An example of a two degree of freedom joint in the plane

is an idealized wheel spinning on ice.

The rotation and translation are completely decoupled.

Kutzbach's equation determines the number of degrees of freedom in a planar system.

M is the number of degrees of freedom, N is the number of links in the system,

J1 is the number of one-degree of freedom joints, and

J2 is the number of two-degree of freedom joints.

Once solved, if M is equal to zero,

the system is fully constrained and can't move.

Another class of constraints are called non-holonomic,

meaning they cannot be described using the position of the system.

These constraints that are not only a function of position can, for

example, be a function of velocity.

In this video, we see that the car wants to move sideways into the parking spot.

However, there's a non-holonomic constraint preventing

any lateral velocity.

The car can however translate in that direction using a parallel parking

manuever.

We are now interested in establishing a relationship between the position of

the robots end effector, X, in its free space, and

all of the machines joint angles, theta.

The forward kinematics are when the joint angles are known and

the position of the end effector is to be determined.

Typically, in robot arms, the joint angles are known as they are instrumented, and

so it is of interest to figure out where the end effector will be

given a set of joint angles.

Inverse kinematics pose the opposite problem where the position

of the end effector is known and a joint angles are to be computed.

In this video, we see a robot arm doing a welding task.

The end effector needs to follow a desired trajectory in ambiance space and so

inverse kinematics must be use back out the joint angles.

These joint angles are then passed to the machine as desired angles for the motor.

Here we see a much simpler revolute-revolute robot arm.

The angle between the first link and the base is theta one and

the angle between the first link and the second link is theta two.

It's important to note that theta two is a relative angle.

Since the position of the end effector, X, is a function, F,

of the joint angles, theta, calculating F represents the forward kinematics.

Some dimensions have been broken out in this image to help solve the problem of

forward kinematics.

Using the same revolute-revolute arm, if the end effector position is known,

inverse kinematics can let us solve for the joint angles.

Since the forward kinematics are known,

this represents a system of non-linear equations.

The inverse kinematics can be found from the forward kinematics

by solving a system of equations.

Generally, this system of equations is non-linear and

can be solved with trigonometric identities.

Simpler cases can be solved in MATLAB by making the desired variables symbolic and

using the solve function.

Solving the inverse kinematics amounts to inverting

the system of equations represented by the function F.

All the previous linkages that we've considered are serial

in that there's only one chain between the ground and the end effector.

Parallel mechanisms have multiple chains connecting the end effector to the ground.

Solving the kinematics of parallel mechanism is much more difficult than for

serial mechanisms.

Here we will go over a concrete example using the minitaur leg.

In this figure we see the minitaur leg with the motor angles theta one and

theta two Illustrated.

We would like to solve the forward kinematics and

so get a solution for X and Y as a function of theta one and theta two.

This task greatly simplified if we use a couple of coordinate changes.

The first coordinate change that will be helpful is to go from theta one and

theta two to alpha and beta, the mean and different coordinates.

Once alpha and beta are computed,

a further coordinate change to polar coordinates help solve the problem.

We now consider the infinitesimal kinematics for these linkages.

This is done by computing the matrix of partial derivatives of the system

known as the Jacobian.

Here we see the Jacobian,

J, computed where the rows represent the system of equations and

the columns represent the partial derivatives of the different joint angles.

By multiplying the Jacobian by DT over DT and rearranging,

we see that the Jacobian describes the relationship between joint velocities and

velocities at the end effector.

Thanks to the principle of virtual work the Jacobian transpose

can be used to relate the forces at the end effector to the torques at the joints.

This Jacobian transpose inverse represents the effective mechanical advantage for

the system.

It scales the torques at the joint into forces at the end effector.

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