mls94-complete[1]

A significantly more di cult problem is that of constructing controllers for robot systems. Although conceptually simple, a controller for a multifingered robot hand must be able to control a very complex system with many degrees of freedom, large amounts of sensory data, and multiple control objectives. A typical hand might have 10–20 actuators, 10–15 constraints, and a state-space of dimension 30 or higher. A control law for such a system might need to run at a control frequency of 500 Hz or more in order to yield acceptable performance. Computing the control torques for such a system in under 2 milliseconds is often impossible if the system is modeled as a single, complex dynamical system.

The di culties in controlling systems with many degrees of freedom have also been noted in the biomechanics literature. The study of human biological motor control mechanisms led the Russian psychologist Bernstein to question how the brain could control a system with so many di erent degrees of freedom interacting in such a complex fashion [41]. Many of these same complexities are also present in robotic systems and limit our ability to use multifingered hands and other robotic systems to their full advantage.

In the remainder of this section, we describe one possible way of structuring controllers which attempts to address some of the di culties inherent in the control of constrained robot systems with many degrees of freedom. We present a set of primitive operations that allow a complex robot controller to be built up in a hierarchical fashion and discuss some of the issues involved in the resulting control structure. The material contained in this section was originally presented in [75], where a more detailed description is given.

Defining robots

We wish to build up complex control laws by utilizing the geometric constraints between the mechanisms which make up the overall system. We will model all mechanisms as a generalized object which we label as a robot. A robot consists of a dynamical system whose equations of motion have the form

M (q)¨q + C(q, q˙)q˙ + N (q, q˙) = F.

The quantities M , C, and N completely parameterize the dynamics of the mechanism.

In addition to the parameters (M, C, N ), a robot also has a set of inputs and outputs. The inputs consist of the desired position of the robot, xd, and the forces to be applied to the robot, Fd. The outputs are the actual position of the mechanism, x, and the measured force F . Some types of robots may not use or define all of these inputs and outputs.

The relationship between the various parameters describing a robot depends on the robot itself. For example, we model an actuated, open-

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chain mechanism using the relationships

F = Fd

M (q)¨q + C(q, q˙)q˙ + N (q, q˙) = F.

Thus, given a desired force to be applied to the robot, the robot will move according to the equations of motion. The desired trajectory input for an uncontrolled robot is ignored. (We will make use of this input later when we attach controllers to robots.)

In addition to modeling actuated mechanisms, a robot can also describe an inanimate object. In this case, all inputs to the robot are ignored and the outputs from the robot provide information about the current position of the object and the forces acting on it, if available. The inertial parameters (M, C, N ) are used as before to model the dynamics of the object.

The utility of defining a generalized object called a robot is that we may define operations which take one or more generalized robots and yield a new generalized robot. We define two such operations below. In order to define the new object, we must define the inertial properties as well as a description of the inputs and outputs. These are typically defined recursively, so that a composite robot queries and commands its children in response to requests for inputs and outputs.

Attaching robots

The first operation which we define is the attach operation, which reflects geometrical constraints between two or more robots. It creates a new robot object from the attributes of its children. Its definition (and name) is motivated by the attachment of a set of fingers to an object, but its use is much more general.

The arguments to the attach operation are a list of robots, which we refer to as fingers, together with a single object which we refer to as the payload. In addition, we are given a constraint between the configuration variables of the fingers and those of the object. For simplicity, we take this constraint to be of the form h(θ, x) = 0 where θ = (θf1 , . . . , θfk ) Rn is the vector of finger joint angles and x Rp is the configuration of the payload.

To construct a new robot, we use the Lagrange-d’Alembert equations to write the dynamics in terms of the payload variables x Rp. Let (Mf , Cf , Nf ) be the (block-diagonal) parameters for the fingers and (Mp, Cp, Np) be those of the payload. The dynamic parameters for the

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constrained robot are given by

˜ −T −1

M := Mp + GJ Mf J G

˜

purposes of exposition. Note that in order to evaluate M at the current configuration, we can query the payload and each of the fingers for their current inertia matrices and then combine these using the constraints.

To read the configuration of the composite robot, we query the state of all the robots in the list of daughter robots and then solve the (holonomic) constraint h(θ, x) = 0 to find the current payload configuration. Alternatively, if the payload is equipped with sensors (perhaps an external camera which tracks the payload), this data can be used instead. A similar computation or measurement can be used to determine the net force on the object, which will consist of contact forces applied by the fingers and external forces applied by the environment.

Commanding the desired position and force on the robot also uses the constraint equations to distribute information to the fingers and the payload. If all fingers are uncontrolled, actuated mechanisms, then the desired forces will be applied to the actuators and the desired position will be ignored.

A diagram illustrating the data flow in a robot constructed by the attach operation is shown in Figure 6.11. In addition to modeling grasp constraints, the attach operation can be used to model other situations, as described in Section 2.3. For example, we can change from joint space to workspace coordinates or add variables parameterizing the internal motion of a redundant robot.

Controlling robots

The control operation is responsible for assigning a controller to a robot. It is also responsible for creating a new robot with attributes that properly represent the controlled robot. The attributes of the created robot are completely determined by the individual controller. For most controllers, the current state of the controlled robot is equivalent to the current state of the uncontrolled robot. Sending a desired trajectory to a controlled robot would cause the controller to bu er the data and attempt to follow that trajectory. A controlled robot is illustrated in Figure 6.12.

˜ ˜ ˜

The dynamic attributes M , C, and N for the newly created robot are determined by the controller. At one extreme, a controller which compensates for the inertia of the robot would set the dynamic attributes of the controlled robot to zero. This does not imply that the robot is

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Figure 6.13: Hierarchical control scheme for a human finger. (Figure courtesy of D. Curtis Deno)

creates a robot object, it can be inserted at any level in the description of a constrained robot system. Thus, we can easily define hierarchical controllers whose structure mirrors the geometric structure of the system. We illustrate this in the following examples.

Example 6.6. Biological motor control

Figure 6.13 shows a hierarchical control scheme for a human finger. At the highest level, the brain is represented as sensory and motor cortex (where sensory information is perceived and conscious motor commands originate) and brainstem and cerebellar structures (where motor commands are coordinated and sent down the spinal cord). A pair of fingers forms a composite system for grasping which is shown integrated at the

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Figure 6.14: Planar two-fingered hand and a hierarchical control law.

level of the spinal cord. The muscles and sensory organs of each finger form low-level spinal reflex loops. These low-level loops respond more quickly to disturbances than sensory motor pathways which travel to the brain and back. Brain and spinal feedback controllers are represented by double-lined boxes.

The block diagram portion of Figure 6.13 is a (biological) example of a robot system built using the operations described above. Starting from the bottom: two fingers (robots) are defined; each finger is controlled by muscle tension/sti ness and spinal reflexes; the fingers are attached to form a composite hand; the brainstem and cerebellum help control and coordinate motor commands and sensory information; and finally, at the level of the cortex, the fingers are thought of as a pincer which engages in high-level tasks such as picking.

Example 6.7. Hierarchical control of a two-fingered planar hand

As a second, more practical example, consider the planar hand shown in Figure 6.14. The parameters describing the dynamics of the fingers and the object, as well as the constraints between them, are easily computed and are given in earlier examples. We can build a hierarchical controller which is similar to the biological controller described in the previous example. This control structure is shown graphically in Figure 6.15.

At the lowest level, we use simple PD control laws attached directly to the individual fingers. These PD controllers mimic the sti ness provided by muscle coactivation in a biological system. Additionally, controllers at this level might be used to represent spinal reflex actions. At a somewhat higher level, the fingers are attached and considered as a single unit with relatively complicated dynamic attributes and Cartesian configuration.

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box trajectory

Computed

torque

Grasping constraint

Finger kinematics

PD PD

Figure 6.15: A hierarchical controller for multifingered grasping.

At this point, we employ a feedforward controller (computed torque with no error correction) to simplify these dynamic properties, as viewed by higher levels of the brain. With respect to these higher levels, the two fingers appear to be two Cartesian force generators, represented as a single composite robot.

Up to this point, the representation and control strategies do not explicitly involve the box, a payload object. The force generators are next attached to the box, yielding a robot with the dynamic properties of the box, but capable of motion due to the actuation in the fingers. Finally, we use a computed torque controller at the very highest level to allow us to command motions of the box without worrying about the details of muscle actuation. By this controller, we simulate the actions of the cerebellum and brainstem to coordinate motion and correct for errors.

It is helpful to illustrate the flow of information to the highest level control law. In the evaluation of the current box configuration and tra-

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jectory, xb and x˙ b, the following sequence of actions occurs:

Hand: asks for current state, xb and x˙ b

Finger: ask for current state, xf and x˙ f

When we write a set of hand forces, a similar chain of events occurs. The structure in Figure 6.15 also has interesting properties from a

more traditional control viewpoint. The low-level PD controllers can be run at high servo rates (due to their simplicity) and allow us to tune the response of the system to reject high-frequency disturbances. The Cartesian feedforward controller permits a distribution of the calculation of nonlinear compensation terms at various levels, lending itself to multiprocessor implementation. Finally, using a computed torque controller at the highest level gives the flexibility of performing the controller design in the task space and results in a system with linear error dynamics.

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6Summary

The following are the key concepts covered in this chapter:

1.The dynamics of a mechanical system with Lagrangian L(q, q˙), subject to a set of Pfa an constraints of the form

where λ Rk is the vector of Lagrange multipliers. The values of the Lagrange multipliers are given by

−1 T −1 −1 − − ˙

λ = (AM A ) AM (F Cq˙ N ) + Aq˙ .

2.The Lagrange-d’Alembert formulation of the dynamics represents

the motion of the system by projecting the equations of motion onto the subspace of allowable motions. If q = (q1, q2) R(n−k)×k and the constraints have the form

In the special case that the constraint is integrable, these equations agree with those obtained by substituting the constraint into the Lagrangian and then using the unconstrained version of Lagrange’s equations.

3.The dynamics for a multifingered robot hand with joint variables θ Rn and (local) object variables x Rp, subject to the grasp

M (q)¨x + C(q, q˙)x˙ + N (q, q˙) = F, where q = (θ, x) and

˜ −T −1 T

M = Mo + GJh Mf Jh G

˜ −T −1 T d

C = Co + GJh Cf Jh G + Mf dt

˜ −T

N = No + GJh Nf F = GJh−T τ.

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These same equations can be applied to a large number of other robotic systems by choosing G and Jh appropriately.

4.For redundant and/or nonmanipulable robot systems, the hand Jacobian is not invertible, resulting in a more complicated derivation of the equations of motion. For redundant systems, the constraints can be extended to the form

where the rows of Kh span the null space of Jh, and vN represents the internal motions of the system. For nonmanipulable systems, we choose a matrix H which spans the space of allowable object trajectories and write the constraints as

where x˙ = H(q)w represents the object velocity. In both the redundant and nonmanipulable cases, the augmented form of the constraints can be used to derive the equations of motion and put them into the standard form given above.

5.The kinematics of tendon-driven systems are described in terms of a set of extension functions, hi : Q → R, which measures the displacement of the tendon as a function of the joint angles of the system. If a vector of tendon forces f Rk is applied at the end of the tendons, the resulting joint torques are given by

τ= P (θ)f,

where P (θ) Rn×p is the coupling matrix:

P (θ) = ∂h∂θ T (θ).

A tendon-system is said to be force-closure at a point θ if for every vector of joint torques, τ , there exists a set of tendon forces which will generate those torques.

6. The equations of motion for a constrained robot system are de-

correctly defined, the quantities satisfy the following properties:

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