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In Chapters 4, 5, and 6, we studied the forward kinematics, velocity kinematics and statics,

Â and inverse kinematics of open-chain robots.

Â In Chapter 7, and in this single video, I am going to cover all of these topics for

Â closed-chain robots, without going into great detail.

Â Kinematics and statics are generally more complicated for closed-chain robots, because

Â there is such a wide variety of design possibilities.

Â The configuration space of closed-chain robots can be quite complex, since they must satisfy

Â a number of loop-closure equations.

Â There are classes of singularities that don't exist for open-chain robots, and the choice

Â of which joints to actuate and which to leave passive can affect the singularities that

Â occur.

Â Oftentimes the analysis of these robots is based on symmetries and insight into the specific

Â structure of the mechanism.

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

Â The study of closed-chain robots is an active research area, and this chapter just skims

Â the surface.

Â Let's start by looking at some examples.

Â The first example is a 4-degree-of-freedom robot arm.

Â The end-effector moves in x, y, and z, and it rotates about a vertical axis.

Â Although it looks similar to an open-chain robot, it is a closed chain due to the parallelogram-type

Â linkage.

Â The next example is a 4-degree-of-freedom Delta robot.

Â The end-effector moves in x, y, and z, and it rotates about a vertical axis.

Â There are also 3-degree-of-freedom Delta robots that eliminate the rotational motion.

Â The final example is the Stewart platform, which moves with the full 6 degrees of freedom

Â of a rigid body.

Â Each of the 6 legs is actuated by a prismatic joint.

Â At one end of each leg is a spherical joint while the other end has a spherical or universal

Â joint.

Â The Stewart platform is popular for applications like aircraft simulators, since it can move

Â the virtual cockpit with all 6 degrees of freedom.

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

Â A parallel robot is a specific type of closed chain which consists of a moving platform

Â attached to a base through a set of actuated legs.

Â For the rest of this video, I will focus on parallel robots.

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

Â For an open-chain robot, typically each joint has a motor driving it.

Â For parallel robots, many of the joints are unactuated.

Â Open-chain robots tend to have a large workspace, since each extra joint adds to the possible

Â motion of the end-effector.

Â Parallel robots tend to have a small workspace, since each leg in parallel places constraints

Â on the motion of the platform.

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

Â robots tend to be relatively weak.

Â Also, flexibility at the joints and links tends to add.

Â Parallel robots, on the other hand, tend to be stiff and strong, since the end-effector

Â force is distributed among the legs.

Â As we saw in chapter 4, the forward kinematics mapping joint values to end-effector configurations

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

Â On the other hand, there may be multiple solutions to the forward kinematics for parallel robots,

Â and finding them can be challenging.

Â Finally, as we saw in chapter 6, solving the inverse kinematics for an open-chain robot

Â can be tricky.

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

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

Â To solidify our understanding of these characteristics, let's use the Stewart platform as an example.

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

Â The configuration of the {b} frame relative to the {s} frame is T_sb-of-theta, where theta

Â is the vector of joint variables representing the leg lengths.

Â For the ith leg, theta_i is the length of the leg.

Â a_is is the vector from the {s}-frame to leg i's joint at the base, measured in the {s}-frame,

Â and b_ib is the vector from the {b}-frame to the top joint of leg i, measured in the

Â {b}-frame.

Â We can transform b_ib to the {s}-frame by premultiplying by the desired end-effector

Â configuration T_sb, provided we represent the vectors in homogeneous coordinates.

Â Now we can calculate the prismatic joint value theta_i as the distance between b_is and a_is.

Â Inverse kinematics is easy for the Stewart platform.

Â If the legs of the parallel robot are more general open chains, then we have to solve

Â an inverse kinematics problem for each leg.

Â Next let's address the inverse velocity kinematics mapping the end-effector twist to the joint

Â velocities.

Â Let v-hat_i be the unit 3-vector aligned with the direction of positive motion of the i-th

Â axis.

Â Skipping the straightforward derivation, we can define a screw axis V_i, expressed in

Â the {s}-frame, with the linear component v-hat_i and the angular component a_is cross v-hat_i.

Â Then the joint velocity theta-dot_i is equal to the screw axis V_i dotted with the spatial

Â twist V_s; this calculates the component of V_s along the joint axis.

Â Repeating this analysis for all the legs, we can write the ith row of the inverse of

Â the space Jacobian, or J_s-inverse, as the screw axis V_i-transpose.

Â Now if the Jacobian-inverse is invertible, we have the velocity kinematics and statics

Â in the {s} frame: the spatial twist V_s equals J_s times theta-dot and the joint forces tau

Â equals J_s-transpose times F_s, the wrench applied by the end-effector.

Â One of the difficulties of analyzing closed-chain robots, however, is understanding all the

Â possible singularities where the Jacobian is not invertible.

Â Let's consider a simpler robot, the 3-by-RPR parallel mechanism, which is the planar analog

Â of the Stewart platform.

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

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

Â If we put the robot at this configuration, it is at a singularity.

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

Â either rotate counterclockwise or clockwise, and we cannot predict which.

Â Closed-chains can be subject to several types of singularities, described in detail in the

Â book, some of which have no analogs in open-chain robots.

Â Examples include configuration-space singularities, actuator singularities, and end-effector singularities.

Â Some of these singularities occur at configurations where the constraint Jacobian, which is the

Â matrix of derivatives of the loop-closure equations with respect to the passive and

Â actuated joint variables, loses rank.

Â Lastly, we address the forward kinematics problem for closed-chains, which was the first

Â problem we addressed for open chains.

Â The forward kinematics problem often involves solving one or more complex nonlinear equations,

Â and in general the forward kinematics has multiple possible solutions.

Â The 3-by-RPR robot can have up to 6 possible end-effector configurations given a set of

Â prismatic joint extensions.

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

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

Â For a given set of leg extensions, usually there are far fewer real solutions.

Â In practice, it is common to use iterative numerical methods with a nearby solution as

Â an initial guess, similar to the Newton-Raphson method we developed for the inverse kinematics

Â of open chains.

Â In this video I have given you a quick summary of the topics of Chapter 7, which itself is

Â a quick summary of the kinematic analysis of closed-chain robots.

Â The design and analysis of closed-chain robots is an active research field, but Chapter 7

Â should give you a good idea of some of the key issues.

Â So, as Chapter 7 concludes, so does our kinematic and static analysis of robots.

Â In Chapter 8 we will begin our study of robot dynamics, which governs how a robot moves

Â when forces and torques are applied at joints.

Â This will be our springboard to advanced topics, like the design of time-optimal trajectories

Â and controllers for robots.

Â