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In chapters four, five and six,

we studied the forward kinematics,

velocity kinematics, and statics and inverse kinematics of open chain robots.

In chapter seven and in this single video,

I'm going to cover all of these topics for

closed-chain robots without going in the great detail.

Kinematics and statics are generally more complicated for

close chain robots because there's such a wide variety of design possibilities.

The configuration space of closed-chain robots can be quite

complex since they must satisfy a number of loop closure equations.

There are classes of singularities that don't

exist for open-chain robots and the choice of

which joints to actuate and which to leave

passive can affect the singularities that occur.

Often times the analysis of these robots is based on

symmetries and insight into the specific structure of the mechanism.

In this chapter, we take an example-based to look at some of these issues.

The study of closed-chain robots is

an active research area and this chapter just skims the surface.

Let's start by looking at some examples.

The first example is a 4-dof robot arm.

The end-effector moves in X, Y,

and Z and it rotates about a vertical axis.

Although it looks similar to an open-chain robot,

it is a closed-chain due to the parallelogram type linkage.

The next example is a 4-dof Delta robot.

The end-effector moves in X, Y,

and Z and it rotates about a vertical axis.

There are also 3-dof Delta robots that eliminate the rotational motion.

The final example is the Stewart platform which

moves with the full 6-dof of a rigid body.

Each of the six legs is actuated by a prismatic joint.

At one end of each leg is a spherical joint while

the other end has a spherical or universal joint.

The Stewart platform is popular for applications like

aircraft simulators since it can move

the virtual cockpit with all six degrees of freedom.

The Delta robot and the Stewart platform are examples of parallel robots.

The parallel robots is a specific type of closed-chain which consists of

a moving platform attached to a base through a set of actuated legs.

For the rest of this video,

I'll focus on parallel robots.

Let's summarize some typical characteristics of open chain and parallel robots.

For an open chain robot,

typically each joint has a motor driving it.

For parallel robots, many of the joints are unactuated.

Open chain robots tend to have a large workspace since

each extra joint adds to the possible motion of the end-effector.

Parallel robots tend to have a small workspace since each leg

in parallel places constraints on the motion of the platform.

Each joint of an open chain robot has to support all of the end-effector force,

so open chain robots tend to be relatively weak.

Also, flexibility at the joints and lengths tend to add.

Parallel robots on the other hand tend to be stiff and

strong since the end-effector force is distributed among the legs.

As we saw in chapter four,

the forward kinematics mapping joint values to end-effector configurations is

relatively easy to evaluate for open chain robots using the product of exponentials.

On the other hand, there may be multiple solutions to

the forward kinematics for parallel robots and finding them can be challenging.

Finally, as we saw in chapter six,

solving inverse kinematics for an open chain robot can be tricky.

There may be multiple solutions and numerical methods may be required to find them.

The inverse kinematics of a parallel robot is sometimes straightforward as we will see.

To solidify our understanding of these characteristics,

let's use the Stewart platform as an example.

The fixed frame is s and the end-effector frame is b.

The configuration of the b frame relative to the s frame is Tsb of theta,

where theta is the vector of joint variables representing the leg lengths.

For the f leg theta I is the length of the leg.

Ais is the vector from the s frame to leg i's joint at the base measured in

the s frame and bib is the vector from

the b frame to the top joint of the leg i measured in the b frame.

We can transform bib to the s frame by

pre-multiplying by the desired end-effector configuration Tsb,

provided we represent the vectors in homogeneous coordinates.

Now, we can calculate the prismatic joint value theta

i as the distance between bis and ais.

Inverse kinematics is easy for the Stewart platform.

If the legs of the parallel robot are

more general open chains then we have to

solve an inverse kinematics problem for each leg.

Next, let's address the inverse velocity kinematics

mapping the end-effector twist to the joint velocities.

Let the v hat i be the unit three vector

aligned with the direction of positive motion of the F-axis.

Skipping the straightforward derivation,

we can define a screw axis Vi expressed in the s frame with

the linear component v head i and the angular component ais cross v hat i.

Then the joint velocity theta.i is equal to

the screw axis Vi dotted with the special twist Vs.

This calculates the component Vs along the joint axis.

Repeating this analysis for all the legs,

we can write the i throw of the inverse of the space Jacobian or

Js inverse as the screw axis Vi transpose.

Now, if the Jacobian inverse is invertible.

We have the velocity kinematics and statics in the s frame.

The spatial twist Vs equals Js times theta dot.

And the joint forces tau equals Js

transpose times Fs the wrench applied by the end-effector.

One of the difficulties of analyzing closed-chain robots however is

understanding all the possible singularities when the Jacobian is not invertible.

Let's consider a simpler robot,

the 3 by RPR parallel mechanism which is the planar analog of the Stewart platform.

The platform moves in all three planar degrees of freedom and is driven by three legs.

Each leg has two unactuated revolute joints and one actuated prismatic joint.

If we put the robot at this configuration,

it is at a singularity.

From this configuration, if we extend the legs at an equal rate,

the platform could either rotate

counterclockwise or clockwise and we can't predict which.

Closed chains can be subject to several types of

singularities described in detail in the book,

some of which have no analogues in open chain robots.

Examples include configuration space singularities,

actuators singularities, and end-effector singularities.

Some of these singularities occurred configurations were the constraint Jacobian,

which is the matrix of derivatives of the loop closure equations with

respect to the passive unactuated joint variables loses rank.

Lastly, we address the forward kinematics problem for closed chains,

which is the first problem we address for open chains.

The forward kinematics problem often involves solving

one or more complex non-linear equations and in

general the forward kinematics has multiple possible solutions.

The 3 x RPR robot can have

up to six possible end-effector configurations given a set of prismatic joint extensions.

This figure shows two possible solutions when all joint extensions are equal.

The 6-dof Stewart platform can have up to 40 solutions for a given set of leg extensions.

For a given set of leg extensions, usually,

there are far fewer real solutions.

In practice, it is common to use iterative numerical methods with

a nearby solution as an initial guess similar to the Newton-Raphson method,

we developed for the inverse kinematics of open chains.

In this video, I have given you a quick summary of the topics of chapter seven,

which itself is a quick summary of the kinematic analysis of closed-chain robots.

The design and analysis of closed-chain robots is an active research field.

But chapter seven should give you a good idea of some of the key issues.

So, as chapter seven concludes,

so does our kinematic and static analysis of robots.

In Chapter 8, we will begin our study of robot dynamics,

which governs how a robot moves when forces and torques are applied at joints.

This will be our springboard to advanced topics like

the design of time optimal trajectories and controllers for robots.