Force Control470 EXAMPLE 9.2–1: Task Space Formulation for a Slanted Surface We want to find the manipulator dynamics for the Cartesian manipulator system (i.e., both joints are prismatic) given in Figure 9.2.5 and to decompose the forces exerted on the surface into a normal force and a tangent force. First, the motion portion of the dynamics can easily be determined when the robot is not constrained by the surface. After removing the surface and the interaction forces f 1 , and f 2 , the manipulator dynamics can be shown to be (1) where and F(q) is the 2×1 vector that models the friction as discussed in Chapter 2. To account for the interaction forces, let x be the 2×1 task space vector defined by (2) where u and v define a fixed coordinate system such that u represents the normal distance to the surface, and v represents the tangent distance along the surface. As in (9.2.9), the task space coordinates can be expressed in terms of the joint space coordinates by (3) where h(q) is found from the geometry of the problem to be (4) Copyright © 2004 by Marcel Dekker, Inc. 9.2 Stiffness Control 471 The task space Jacobian matrix is found from (9.2.11) by utilizing the fact that T is the identity matrix for this problem because we do not have to concern ourselves with any end-effector angles of orientation. That is, J(q) is given as (5) Following (9.2.14), the robot manipulator equation is given by (6) where It is important to realize that the normal force (i.e., f 1 ) and the tangent force (i.e., f 2 ) are drawn in the direction of the task space coordinate system given by (2) (see Fig. 9.2.5). EXAMPLE 9.2–2: Task Space Formulation for an Elliptical Surface We wish to find the manipulator dynamics for the Cartesian manipulator system given in Figure 9.2.6 and to decompose the forces exerted on the surface into a normal force and a tangent force. The motion portion of the dynamics is the same as in Example 9.2.1; however, due to the change in the environmental surface, a new task space coordinate system must be defined. Specifically, let x be the 2×1 task space vector defined by (1) where u and v define a rotating coordinate system such that u represents the normal distance to the surface and v represents the tangent distance along the surface. As in (9.2.9), the task space coordinates can be expressed in terms of the joint space coordinates by (2) Copyright © 2004 by Marcel Dekker, Inc. 9.2 Stiffness Control 473 The partial derivative of (6) with respect to q 2 divided by the length of the vector yields a unit vector (i.e., v) that is always tangent to the surface. That is, v is given by (7) where By using (5) again, the expression for v can be simplified to yield (8) where Because the vectors u and v must be orthogonal (i.e., u.v=0), via (8) and the geometry of the problem, we know that (9) Substituting (4), (8), and (9) into (3) yields (10) The task space Jacobian matrix is found from (9.2.11) by utilizing the fact that T is the identity matrix. That is, J(q) is given as (11) where Copyright © 2004 by Marcel Dekker, Inc. Force Control474 Following (9.2.14), the robot manipulator equation is given by (12) where , M, q, G, f, and F(q) are as defined in Example 9.2.1. It is important to note that the normal force (i.e., f 1 ) and the tangent force (i.e., f 2 ) are drawn in the direction of the task space coordinate system given by (1) (see Fig. 9.2.6). Stiffness Control of an N-Link Manipulator Now that we have the robot manipulator dynamics in a form which includes the environmental interaction forces, the stiffness controller for the n-link robot manipulator can be formulated. As before, the force exerted on the environment is defined as (9.2.17) where K e is an n×n diagonal, positive semi-definite, constant matrix used to denote the environmental stiffness, and x e is an n×1 vector measured in task space that is used to denote the static location of the environment. Note that if the manipulator is not constrained in a particular task space direction, the corresponding diagonal element of the matrix K e is assumed to be zero. Also, the environmental surface friction is typically neglected in the stiffness control formulation. The multidimensional stiffness controller is the PD-type controller (9.2.18) where K v and K p are n×n diagonal, constant, positive-definite matrices and the task space tracking error is defined as Copyright © 2004 by Marcel Dekker, Inc. 9.2 Stiffness Control 475 As before, x d is used to denote the desired constant end-effector position that we wish to move the robot manipulator to; however, x d is now an n×1 vector. Substituting (9.2.17) and (9.2.18) into (9.2.14) yields the closed-loop dynamics (9.2.19) To analyze the stability of the system given by (9.2.19), we utilize the Lyapunov-like function (9.2.20) Differentiating (9.2.20) with respect to time and utilizing (9.2.10) yields (9.2.21) Note that in (9.2.21) we have used the fact that x e and x d are constant and that the transpose of either a scalar function or a diagonal matrix is equal to that function or matrix, respectively. Substituting (9.2.19) into (9.2.21) and utilizing (9.2.10) yields (9.2.22) Applying the skew-symmetric property (see Chapter 2) to (9.2.22) yields (9.2.23) which is nonpositive. Since the matrices J(q) and hence J T (q)K v J(q) are nonsingular, V can only remain zero along trajectories where q=0 and hence q=0 (see LaSalle’s theorem in Chapter 1). Substituting q=0 and q=0 into (9.2.19) and utilizing (9.2.10) yields (9.2.24) or, equivalently, (9.2.25) where the subscript i is used to denote the ith component of the vectors x, x d , x e , and the ith diagonal element of the matrices K p and K e . Copyright © 2004 by Marcel Dekker, Inc. Force Control476 The stability analysis above can be interpreted to mean that the robot manipulator will stop moving when the task space coordinates are given by (9.2.25). That is, the final position or steady-state position of the end effector is given by (9.2.25), which in the single-degree-of-freedom case is equivalently given by (9.2.6). To obtain the ith component of the steady-state force exerted on the environment, we substitute (9.2.25) into the ith component of (9.2.17) to yield (9.2.26) Thus the steady-state force exerted by the end effector on the environment is given by (9.2.26), which in the single-degree-of-freedom case is equivalently given by (9.2.7). As in the single-degree-of-freedom case, we assume that K ei is much larger than K pi for the task space directions that are to be force controlled. That is, the steady-state force in (9.2.26) can be approximated by Table 9.2.1: Stiffness Controller (9.2.27) Copyright © 2004 by Marcel Dekker, Inc. 9.2 Stiffness Control 477 therefore, K pi can interpreted as specifying the stiffness of the manipulator in these task space directions. If the manipulator is not constrained in a task space direction, the corresponding stiffness constant K ei is equal to zero. Substituting K ei =0 into (9.2.25) yields (9.2.28) This means that for the nonconstrained task space directions, we obtain set- point control; therefore, in steady state the desired position set point is reached. The stiffness controller along with the corresponding stability result are both summarized in Table 9.2.1. We now illustrate the concept of stiffness control with an example. EXAMPLE 9.2–3: Stiffness Controller for a Cartesian Manipulator We want to design and simulate a stiffness controller for the robot manipulator system given in Figure 9.2.5. The control objective is to move the end effector to a desired final position of v d =3 m while exerting a final desired normal force of f d1 =2 N. We neglect the surface friction (i.e., f 2 ) and joint friction, and assume that the normal force (i.e., f 1 ) satisfies the relationship (1) where and k e =1000 N/m. The robot link masses are assumed to be unity, and the initial end-effector position is given by (2) To accomplish the control objective, the stiffness controller from Table 9.2.1 is given by (3) where Copyright © 2004 by Marcel Dekker, Inc. Force Control478 , J, G, and x are as defined in Example 9.2.1, u d is defined as the desired normal position, and the gain matrices K v and K p have been taken to be K v =k v I and K p =k p I. For this example we select k v = k p =10, which will guarantee that k p << k e as required in the stiffness control formulation. To satisfy the control objective that f d1 =2 N, we utilize (9.2.27) to determine the desired normal position. Specifically, substituting the values of f d1 , k e , and u e into (4) yields The simulation of the stiffness controller given by (3) for the robot manipulator system (Figure 9.2.5) is given in Figure 9.2.7. As indicated by the simulation, the desired tangential position and normal force are reached in about 4 s. 9.3 Hybrid Position/Force Control A major disadvantage of the stiffness controller given in Section 9.2 is that it can only be used for set-point control; in other words, the desired end effector manipulator position and the desired force exerted on the environment must be constant. In many robotic applications, such as grinding, the end effector must track a desired positional trajectory along the object surface while Figure 9.2.7: Simulation of stiffness controller. Copyright © 2004 by Marcel Dekker, Inc. 479 tracking a desired force trajectory exerted onto the object surface. In this type of application, a stiffness controller will not perform adequately; therefore, another control approach must be utilized. The so-called hybrid position/force controller [Chae et al. 1988] and [Raibert and Craig 1981] can be used for tracking position and force trajectories simultaneously. The basic concept of the hybrid position/force controller is to decouple the position and force control problems into subtasks via a task space formulation. As we have seen, the task space formulation is valuable in determining which directions should be force or position controlled. That is, the position and force control subtasks are easily determined from the task space formulation. After the control subtasks have been identified, separate position and force controllers can then be developed. Hybrid Position/Force Control of a Cartesian Two-Link Arm To illustrate this concept of hybrid position/force control, consider the robot manipulator system given by Figure 9.3.1. For this application, the position along the surface and the normal force exerted on the surface should both be controlled; therefore, one must determine which variables should be force controlled and which should be position controlled. 9.3 Hybrid Position/Force Control Figure 9.3.1: Manipulator moving along perpendicular surface. Copyright © 2004 by Marcel Dekker, Inc. [...]... Position/Force Control of an N-Link Manipulator The hybrid position/force controller given in the preceding section can easily be extended to the multidegree case by using the task space formulation concept Specifically, one can develop a feedback-linearizing control that will globally linearize the robot manipulator equation and then develop linear controllers to track the desired force and position... position and force control must be formulated For position control [Anderson and Spong 1988], the relationship between force and velocity is modeled by (9.4.5) where xd represents the input velocity of the manipulator at the environmental contact point and Zm(s) represents the manipulator impedance Copyright © 2004 by Marcel Dekker, Inc 9.4 Hybrid Impedance Control 493 As we will show subsequently, the manipulator. .. the lefthand side of (9.4.15) as (9.4.16) where Zpmi is the ith position-controlled manipulator impedance Therefore, equating (9.4.16) and (9.4.15) gives the ith position controller (9.4.17) where L-1 is used to represent the inverse Laplace transform operation Continuing with the separation of position and force control designs, we use (9.4 .13) to define the equations that are to be force controlled... position/force controller is a feedback linearizing controller; therefore, exact knowledge of the robot dynamics is required The reduced state controller [McClamroch and Wang 1988] is (9.5.16) where Kv and Kp are diagonal, positive-definite (n-p)×(n-p) matrices, and Kf is a diagonal, positive-definite p×p matrix To determine the type of stability for the position error and force tracking error, we substitute (9.5.16)... (9.4.20) (9.4.21) where Zfmj is the jth force-controlled manipulator impedance Therefore, equating (9.4.21) and (9.4.19) gives the jth force controller The overall “hybrid” impedance control strategy is obtained by using (9.4.12) in conjunction with (9.4.17) and (9.4.20) This hybrid impedance control strategy is summarized in Table 9.4.1 Note that a higher-level controller would be used to select the components... Inc 482 Force Control (9.3.12) where fd1 represents the desired normal force that is to be exerted on the environment Similar to the position controller, the force controller will be the computed-torque controller (9.3 .13) where (9.3.14) with kNv and kNp being positive control gains Substituting (9.3 .13) into (9.3.11) gives the force tracking error system, (9.3.15) Using the fact kNv and kNp are positive... by (6) and (7) , and ke1=1000 where The simulation of the hybrid position/force controller given by (3), (6), and (7) for the robotic manipulator system (Figure 9.2.5) is given in Figure 9.3.2 The controller gains were selected as As indicated by the simulation, the position and force tracking error go to zero in about 4 s Implementation Issues After reexamining the hybrid position/force controller,... the subscript j denotes the jth force-controlled task space variable, and the subscript f denotes force control The associated environmental forces in the force-controlled task space directions are denoted by ffj Assuming zero initial conditions, the Laplace transform of (9.4.18) can be written as: (9.4.19) From the force control model given in (9.4.8), the left-hand side of (9.4.19) can also be written... inertial manipulator impedances The discussion above can be summarized by the following duality principle DUALITY PRINCIPLE Capacitive environments are force controlled with noncapacitive manipulator impedances, inertial environments are position controlled with noninertial manipulator impedances, and resistive environments are force controlled with inertial manipulator impedances or position controlled... 2004 by Marcel Dekker, Inc 498 Force Control (2) where he, be, de, and ke are all positive scalar constants From Table 9.4.1, the hybrid impedance controller is given by (3) where a is a 2×1 vector representing the separate position and force control strategies, and , J, G, and f are as defined in Example 9.2.1 The torque controller given by (3) decouples the robot dynamics in the task space as follows: . Specifically, one can develop a feedback-linearizing control that will globally linearize the robot manipulator equation and then develop linear controllers to track the desired force and position. of application, a stiffness controller will not perform adequately; therefore, another control approach must be utilized. The so-called hybrid position/force controller [Chae et al. 1988] and [Raibert. typically neglected in the stiffness control formulation. The multidimensional stiffness controller is the PD-type controller (9.2.18) where K v and K p are n×n diagonal, constant, positive-definite