AN EFFICIENT GRADIENT PROJECTION OPTIMIZATION SCHEME FOR A SEVEN-DEGREE-OF-FREEDOM REDUNDANT ROBOT WITH SPHERICAL WRIST"

Transcription

1 AN EFFICIENT GRADIENT PROJECTION OPTIMIZATION SCHEME FOR A SEVEN-DEGREE-OF-FREEDOM REDUNDANT ROBOT WITH SPHERICAL WRIST" R. V. Dubey,t J. A. Euler,t and S. M. Babcocks Center for Engineering Systems Advanced Research Oak Ridge National Laboratory Oak Ridge, Tennessee Abstract A computationally efficient kinematic control scheme is presented for a seven-degree-of-freedom redundant robot with spherical wrist. This scheme uses a gradient projection optimization method, which eliminates the need to determine the generalized inverse of the Jacobian when solving for joint velocities for given Cartesian end-effector velocities. Closed-form solutions are obtained for joint velocities using this approach. The application of this scheme to the seven-degree-of-freedomanipulator at the Center for Engineering Systems Advanced Research (CESAR) is described. 1. Introduction The robots presently used in industry possess six joints or degrees of freedom. Six is the minimum number of joints required to reach an arbitrary position and an orientation of the robot end-effector within its working region. However, robots with six joints are limited in their ability to follow an arbitrarily specified end-effector path. A significant portion of a six-joint robot's workspace is occupied by singular configurations. In a singular configuration the robot end-effector is unable to move or rotate in certain directions. The presence of obstacles may also constrain the movement of a robot in many situations. Similarly, limited range of motion of the joints and torque limits on motors also restrict the movement of a six degree-offreedom robot. Therefore, it is highly desirable for a general purpose robotic manipulacor to have more than six degrees of freedom. Additional joints, referred to as redundant joints, can be added to robots to overcome these problems. These joints can be brought into play to avoid the above problems and to improve the performance of the robot. It has been suggested that future *Research sponsored by the Engineering Research Program of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. tdepartment of Mechanical and Aerospace Engineering, The University of Tennessee, Knoxville, TN Instrumentation and Controls Division. general purpose robots should have at least seven degrees of freedom.2 Several joints of a robotic manipulator move simultaneously at time-varying rates in order to follow a specified end-effector path. The desired end-effector motion must be resolved into the necessary joint motions. In case of a six degree-of-freedom manipulator, the joint velocities required to achieve the desired translational and rotational end-effector motion are unique. However, a redundant robot is one that has more joints than the six independent Cartesian variables (translational and rotational velocity components) for the end-effector. Thus, the kinematic linear equations relating unknown joint velocities to specified end-effector velocity components in world coordinates do not have a unique solution-an infinite number of solutions are possible. Thus, joint velocities may be selected to achieve a secondary objective besides the primary objective of moving the endeffector with desired translational and rotational velocities. The secondary objective may be defined in terms of optimizing a performance criterion such as minimizing torque requirements at the joint motors or maximizing the distance of the robot elbow from an obstacle. A number of control schemes for determining the joint trajectories for redundant robots have been developed using both global and local resolution of redundancy. The global redundancy resolution schemes determine a joint trajectory from the complete description of the desired endeffector trajectory. The selection of the joint trajectory is based on the global optimization of a performance criterion such as average kinetic energy. Global schemes generally are very complex and currently are limited to off-line programming as opposed to real-time control. The local redundancy resolution schemes, on the other hand, determine a joint trajectory from the instantaneous joint motion required to follow a desired end-effector trajectory. The joint motions are obtained by satisfying the local optimization of a performance criterion. Even though local schemes may not result in the "best" trajectory, they are most suitable for real-time control in a changing environment. Chang4 proposed a local optimization scheme in which a set of equations are obtained as constraints on joint angle values. This set is in addition to the original set of nonlinear equations relating the end-effector position and orientation to the joint angles. Wampler5 has U.S. Government Work. Not protected by U.S. copyright. 28

2 proposed inverse kinematic functions for robots to resolve redundancy. It has been suggested by researchers to use the least-norm joint velocity solution for resolved rate control as it minimizes the instantaneous joint velocities. Whitney6' ' proposed to minimize the instantaneous kinetic energy along the trajectory by using the weighted least-norm solution, with the inertia matrix as the weighting matrix. Baillieul, Hollerbach, and Brocketts showed that using the least-norm solution alone for joint velocities may generate trajectories that pass arbitrarily close to singular points. The null space of the Jacobian, which corresponds to the self-motion of the robot, has been utilized by many researchers to optimize various performance criteria of the robot. Liegeoisg developed a gradient projection scheme to utilize the null space of the Jacobian to optimize a joint position-dependent scalar performance criterion. Yoshikawa" introduced a performance criterion Jdet(JJT), where J is the Jacobian matrix. A number of other performance criteria for robots have been suggested in the literature. Luh and suggested the use of the trace of (JJT) to study the efficiency and flexibility of redundant robots. Salisbury and Craig13 and Asada and Cro Granito14 used the condition number as a work space quality index. Klein15 suggested that the minimum singular value is a good measure of work space quality as it provides the upper bound of the velocity with which the end-effector can be moved in all directions. Dubey and Luh16-" used the gradient projection method to improve the efficiency, mechanical advantage, and flexibility of the manipulator. This was achieved by improving the quantitative performance measures called Manipulator-Velocity- Ratio and Manipulator-Mechanical-Advantage. Konstantinov, Markov, and Nenchev18 used the null space of the Jacobian to satisfy a secondary criterion in the least-square sense. Hanafusa, Yoshikawa, and Nakamuralg used a similar approach by dividing a task into primary (high priority) and secondary (low priority) tasks and utilizing the null space to achieve the secondary task in the least-square sense. Hollerbach and Sub" developed a similar scheme using dynamics to minimize joint torques in the least-square sense. Baillieulzl proposed an extended Jacobian method, which involves adding an optimization equation to the original set of kinematic equations so as to make extreme a performance criterion. Dubey and Walkerz2 proposed an inverse kinematic scheme for redundant manipulators by dividing the joints into primary and secondary joints and using the secondary joints to avoid singular configurations. When a secondary criterion is satisfied in the least-square sensez*'* in a local optimization scheme, the robot is required to reconfigure itself to the optimum configuration within a sampling period. If the robot is away from the optimum configuration, reconfiguration would require extremely high joint velocities. Because such high speeds are physically not realizable, the result may be highly inaccurate motion. An extended Jacobian schemez1 also requires the robot to move to the optimum configuration at each sampling signal. Extremely high joint velocities are obtained when the extended Jacobian is close to being singular, and a high joint velocity might occur even though the original Jacobian may not be close to being singular. Such a situation is referred to as algorithmic singularity. Optimization of a performance criterion using the gradient projection method tends to move the robot gradually toward the optimum configuration while the end-effector follows a desired trajectory. The gradient projection method determines the direction of the homogeneous solution or self-motion, while the magnitude may be selected to avoid the bounds of joint velocities. This paper presents an efficient gradient projection optimization scheme for a sevendegree-of-freedom redundant robot with a spherical wrist. In Section 2, we discuss the gradient projection method for the Jacobian control or resolved motion control of redundant robots. Section 3 presents an efficient control algorithm using the gradient projection method for a seven-degree-of-freedom robot with spherical wrist. The application of this control scheme to the CESAR seven-degree-of-freedom manipulatorz3 is described in Section 4. Simulation results demonstrating the feasibility and effectiveness of the control scheme are also presented in this section. The last section presents the conclusions of this paper. 2. The Gradient Proiection Method A manipulator with n joints that are used to control m independent variables of the endeffector position and orientation (m s 6) is described by the following kinematic equation: - k = J b, (1) where is an m-dimensional vector of translational and rotational velocities of the end-effector with reference to base coordinates, e is an n-dimensional vector of joint velocities, and J is an m x n Jacobian matrix. Unless otherwise stated, m will be assumed equal to 6, implying that six independent variables are required to describe the end-effector. If J is a square matrix (m = n) and has a rank equal to m (full rank), then the joint velocities required to achieve the desired end-effector motion will be unique and can be evaluated by J-'k. (2) If J is singular or rectangular with m < n, the vector e can be computed by - e = J+& + (I-J+J)~, (3) where J+ is the Moore-Penrose generalized inversez4 of the Jacobian. The matrix I in Eq. (3) is an n x n identity matrix and (I - J'J) ys the null space projection matrix. The vector L is an arbitrary velocity vector. The solution obtained using Eq. (3) is a least-square solution [i.e., it provides a joint velocity vector that minimizes the Euclidean norm of (Je - 2) for a 29

3 given &]. The first term on the right of Eq. (3), J+g, is the least-norm solution. The second term, (I - J+J)& E N(J), the null space of J, is a homogeneous solution which is orthogonal to J'B. The homogeneous solution is referred to as the self-motion of the manipulator and does not cause any end-effector motion. For a desired trajectory, the homogeneous solution should be appropriately selected to improve the performance of the robot. In order to improve a performance criterion H(e) using the gradient projection method, the redundancy is resolved by substituting kvh(s) for & in Eq. (3) and rewriting it as J'i + k(i-j+j) W(e). (4) The coefficient k in Eq. (4) is a real scalar constant, and VH(B), the gradient vector of H@), is described as The scalar constant k is taken to be positive if H(B) is to be maximized and negative if H(8) is to be minimized. A larger value of k will optimize H(B) at a faster rate. However, the maximum allowable value of k is limited by bounds on the joint velocities. A major problem with the gradient projection method as presented in the literature is its complexity when applied to "real-world" robots. If we follow Eq. (4) directly, it is required to determine the pseudo-inverse of the Jacobian and its null space projection matrix in order to determine the least-norm solution and homogeneous solution respectively. As suggested by Klein, ls using the Gaussian elimination method can reduce the computation time to some extent. However, this approach still requires the computation of JJT and the numerical solution of six simultaneous equations with six unknowns. In Section 3 a scheme is presented that eliminates the need to determine the pseudo-inverse of the Jacobian. By use of a geometrical projection method, the problem is reduced to solving two sets of three simultaneous equations for a sevendegree-of-freedom redundant robot with a spherical wrist. This solution allows us to obtain a closed-form solution of the joint velocities. 3. An Efficient Control Scheme We now present an efficient control scheme for determining the joint velocities given by Eq. (4). This scheme is presented for a sevendegree-of-freedom redundant robot with a spherical wrist. Seven is the minimum number of joints required for a robot to be able to avoid internal singularities and obstacles while following a specified end-effector trajectory. A spherical wrist, which is commonly used in industrial robots, allows for simpler kinematic calculafions. Let e E R7 be the joint velocity vector for the seven-degree-of-freedom robot. Suppose in the Cartesian workspace the end-effector velocity vector with reference to the base coordinates is represented by - X - (vl v2 v3 w1 w2 w ~ E ) R6 ~ (6) and has first three translational and last three rotational velocjty components. The joint velocity vector e and the end-effector velocity vector & are related by - X-J$, (7) where J is a 6 x 7 Jacobian matrix. We will assume that the rank of the Jacobian is six, which implies that J is not singular. Thus, it is possible to construct a nonsingular 6 x 6 matrix J* from any six independent columns of the Jacobian. In general, by rearranging the columns of J and the corresponding elements of e in a different order, we can rewrite Eq. (7) as where a is any column vector of the Jacobian such that the remaining six columns form a nonsingular matrix J*. Any joint velocity vector e satisfying Eq. (7) can be written as $p + kbh, (9) where bp E R7 is any.particular solution satisfying Eq. (7), eh E R7 is a homogeneous solution of Eq. (7) satisfying and k is a scalar constant. In order to determine a particular solution ep, we assume that the first element of Bp is zero. The remaining elements can now be solved from Eq. (8). Thus, we obtain the particular solution as A homogeneous solution &, may be obtained by assuming the first element of &, to be equal to one and solving for [gj*]bh-o. This gives the homogeneous solution bh as - [ -:-.la]. Substituting Eqs. (11) and (13) in Eq. (9), we obtain where k is an arbitrary scalar constant. Least-Norm Solution The least-norm solution kin is the joint velocity vector that satisfies Eq. (1) and has minimum Euclidean norm. Using the generallzed inverse J+ of the Jacobian, we can obtain kin as We will now develop an expression of kin in terms of Bp and e,, obtained from Eqs. (11) and (13) respectively. This expression will avoid the need to evaluate the pseudo-inverse of the (15) 30

4 Jacobian. Using Eq. (9), the square of the norm of e can be written as = ]I$ + k&112 = ($ + k&)t($ + k&).(16) Taking the partial derivative of the right-hand side of Eq. (16) with respect to k and setting it equal to zero gives us (bp + kbh)tbh = 0. (17) Solving for the scalar constant k from Eq. (17), we obtain.. ep' eh k-- - 6hT &h (18) By substituting Eq. (18) in Eq. (9), we obtain the expfession for the least-norm joint velocity vector sin as J* is a nonsingular matrix formed by any six independent columns of the Jacobian J, and I. is the remaining column of J. If we assume the wrist to be spherical, the Jacobian of a seven-degree-of-freedom redundant robot can always be partitioned as follows: where all the elements of the 3 x 3 matrix 0 are zero. If we assume that the spherical wrist is not in a singular configuration, the last three columns of J in Eq. (26) will be independent. Including the last three columns of J in J*. we can always partition J* as follows: If uh is defined as a unit vector in the direction of eh, then ernin can be rewritten as where k i n = 6p - (6; ' uh)gh I (20) Thus, the least-norm solution is obtained by subtracting from the particular solution the component of the particular solution in the direction of the homogeneous solution. The ODtimization Scheme We now develop a scheme to determine the joint velocities in Eq. (4) without determining the generalized inverse. Rearranging the terms in Eq. (4), we obtain the following: 6 = J'(i - WVH) + kvh. (22) A suitable selection of k may be based on the hardware limits of joint velocities and heuristics. The second term on the right-hand side of Eq. (22) can be evaluated from Eq. (5). The first term on the right-hand side of Eq. (22) is the least-norm solution of Eq. (7) with & replaced by (& - WVH). Thus, using Eqs. (19), (ll), and (13) and replacing & by (2 - WVH), we can rewrite Eq. (22) as -1 To determine J"(& - WVH) and J* I. in Eq. (24) and Eq. (25) respectively, we solve the following set of linear equations: J*Y-z. - (X - WVH) where y c R6 and z c R6. Setting z.- and z E respectively and solving for - y from Eq. (28) would give us the values J $1 (2 - WVH) and.jilg. Now From Eqs. (24) and (25) we can evaluate 6; and 63; respectively. Since J* can be partitioned as shown in Eq. (27), we can reduce the problem of solving for a six-dimensional vector y in Eq. (28) to the problem of solving for two tfireedimensional vectors, z1 and yz, such that Let z1 c R3 and zz c R3 be such that z - [z1 ZzIT. It can be easily shown that and -1 - Y1 - J,* z1 I " where and 0" = 0 --P [ -1 J* (g-kjvh) I ' *-1 - Yz = J, (zz - J; TI) (32) As will be shown by an example, it is usually possible to get simple closed-form solutions of il and yz from Eqs.- (31) and (32) respectively. The proelem of determining the joint velocities in Eq. ( 4), using the gradient projection method, has been simplified to the problem of solving sets of three simultaneous equations, which can be further reduced to closed-form solutions for joint velocities. The above results can be organized in the form of an algorithm presented below. 31

5 Aleorithm for the Computation of Joint Velocities for Gradient Proiection Method Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Use the present position and orientation of the end-effector and their desired values at the next sampling instant to determine the desired end-effector velocity vector &. Choose a positive scalar constant k if the performance measure H(8) is to be maximized and a negative scalar constant k if H(B) is to be minimized. Determine the elements of w(b) in Eq. (5) for the current joint angle values. Determine using Eqs. (27) through (32) and Eq. (24). Determine et using Eqs. (27) through (32) and Eq. (25). Determine e using Eq. (2?). If the joint velocities 8 in step 6 are within specified bounds, use these values of 8 for command signals of the control loop. If the joint velocities in step 6 are out of bounds, choose a smaller value 0f.k and repeat steps 3 through 6 until 9 is obtained within bounds; then use these values for command signals. 4. An Illustrative Example The above control scheme for optimizing a performance criterion using the gradient projection method was applied to the CESAR manipulator developed at the Center for Engineering Systems Advanced Research of the Oak Ridge National Laboratory. The CESAR manipulator (Fig. 1) is a seven degree-of- freedom manipulator with a spherical wrist. The pitch-yaw-roll spherical wrist is designed in such a way that its singularities occur when the hand is pointing to its sides and not when it is pointing straight out, as is common in many industrial robots. Degrees of freedom of the CESAR manipulator and the coordinate frames referred to in the Denavit-Hartenberg table (Table 1) for this manipulator are shown in Figs. 2 and 3 respectively. Table 1. Denavit-Hartenberg table of link parameters 1 ' d '3 - d3 90 a3 0 '4 0 0 a ' d Note: d, m, d, m, d m, a m, a, m. In order to simplify the calculations we will refer the desired end-effector and wrist velocity vectors to the third coordinate frame E,, y3, z3. This results in a Jacobian that has a muchsimpler form and, thus, is more efficient for computation. Let the desired end-effector velocity referred to the third coordinate frame be given by: 3ih = [3yi ~ = [3vhl vhz vh3 whl Wh, Wh3IT I (33) SHOULDER PITCH VERTICAL AXIS Fig. 1. CESAR research manipulator. Fig. 2. Seven degrees of freedom of CESAR research manipulator. 32

6 where 34 E R3 and 3s e R3 are the translational and rotational hand velocity vectors, respectively, referred to the third coordinate frame. Let the desired wrist velocity vector, projected onto the third coordinate frame, be given by: L % I & [ 3 T 3 T T - [3VWl vw2 vw3 uwl uw2 uw31t 9 (34) where 'v- 6 R3 and R3 are the translational and rotational wrist velocity vectors, respectively, referred to the third coordinate frame. For a given 3&h, the terms of 3& may be obtained by using the following relationships: 3% =3L?h I (35) = 34 - '$ X d7 34, (36) where 3gh is the unit vector zh at the hand (Fig. 2) that is referred to the third coordinate frame; it may be shown to be the following: 3& = s45s6 C6IT, (37) where c45 = cos(e, + e5), sb5 = sin(@, + e5), c6 = cases, and ss = sines. Let 3Jw be the, Jacobian relating the joint velocity vector e = [e, e2,-- -,e7] and the wrist velocity vector 3 f, that: 3. L=~J,$. The Jacobian 3J.. can be shown to be the 1 following: such (38) Fig. 3. Manipulator coordinate system definition. 3Jw = dzczc,+d, s2s3 (d3-a4s4)c3 0 -a4s a4s2s3s4 -d2 s2+a3 s2 s3 (a3+a4c4)c3 0 a4c a4 s2s3c4 d2c2s3-d3s2c3-a3c2 (d3-a4s4)s3 -a3-a4c a4 (s2c3s4-c2c4) '23' -s3 0 -'45 '45'6, (39) c2 0 1 '45 '45'6 '2 '3 c '6 where si = sinei and ci = cosoi for i = 2 to 6, c45 = cos(0, + Q5), and s45 = sin(0, + e5). In order to determine the joint velocities required to follow a desired end-effector velocity and to optimize a given performance criterion using gradient projection method, we first determine the end-effector velocity 3gh E;] referred to the third coordinate frame. Given the end-effector velocity vector kh E R6 in the base coordinates, we can determine 3&h from the following: We can now determine the wrist velocity vector 3. zw = c3x: from Eqs. (35) and ( 36). Consider the case when the second column of the Jacobian is taken to be E and the remaining six columns are independent and form the matrix J*. Using Eqs. (27) through (32) and Eq. (?4) and rearranging the order, the elemenfs of 9: denoted by e;,, for i = 1 to 7, with et2 = 0, may = ; be obtained a's follows: r c 4 a4 s4 3. Eh = 3R0 &h, (40) d2s2 -a3 szs3 d2czc3+d3 s2s3 where 3R, is a 3 x 3 projection matrix given by: -a4 ' 2 '3 4 ' -a4 ' 2 '3 ' 4 c1c2c3 -s1s3 s1c2c3+c1s3 where A = a4 [dz(c2c3c4 - szs4 ) [ 3R0 = -cisz -s1s2 s2zi], (41) C1CZS3+S1C3 s1c2s3 -c1c3 s2s3 where si - sin@, and ci = cosei for i = 1 to 3. + d,s2s3c4 + a3s2s3s4 1 I 33

7 e;, - '451'6 '451'6 1 -x, + [dzc2s3 - d3s2c3 - a3c2 c4 5 x4 I, 1 - s2c,e;l - c2ep*, - e;, (a, + a4c4) I(44) and et5 - x, - s2s3e;l - e;, - C6 e;, (45) In Eqs. (42) through (45), xi for i - 1 to 6 are the ith elements of the vector (& - WVH). The vector i: with elements 0,", i - 1 to 7 may similarly be obtained by sltting its second element = 1, and the remaining elements may be obtained from E s. (42) through (45) by replacing 0,: by e, s and using the ith elements of vector -a for xi, i = 1 to 6. We have obtained computationally effic;ent closed-form solutions for the elements of and.& in Eqs. (42) through (45). The vector e can now be obtained from Eq. (23). Thus, by this approach we have eliminated the need to determine the generalized inverse of the Jacobian or to numerically solve six simultaneous equations with six unknowns. Simulation Results Simulations depicted in Figures 4 through 7 were performed for the seven degree-of-freedom CESAR manipulator using the efficient gradient projection scheme developed above. Two different cases were considered for simulation. In each case the robot end-effector follows a specified straight line trajectory while maintaining its orientation fixed. In case one, a performance criterion H = sin28, is maximized so that the robot avoids running into its pedestal. In the second case, joint angle limits are avoided by optimizing a performance criterion H - cos28, + cos2e5 + sin28, + cos28,. Comparison is made with the cases when only the least-norm joint velocity vector is used and the null space of the Jacobian is not used to optimize a performance criterion. Fig. 4. Obstacle avoidance example - least-norm velocity solution. Fig. 5. Obstacle avoidance example - performance measure optimized. 34

8 Simulation results are presented using a three-dimensional solid model of the robot. In Fig. 4, only the least-norm solution is used to follow the desired trajectory. In this case the forearm runs into the pedestal as the endeffector moves along the desired crajectory. The null space is utilized to maximize the performance criterion H - sin2b, in Fig. 5, and the forearm of the manipulator avoids the pedestal while following the desired end-effector trajectory. When a specified trajectory is followed in Fig. 6 using the least-norm solution, joint 4 reaches its limit, and the robot stops before reaching the desired end point. On the other hand, as shown in Fig. 7, if the null space of the Jacobian is utilized to maximize the performance criterion H = cos2b, + cos2b5 + sin2bs + cos2b,, the robot reaches the desired end point without any of the joint angles reaching a limit. Through simulation we have shown the effectiveness of the gradient projection control scheme developed in this paper. 5. Conclusions An efficient control scheme using the gradient projection method was developed for a sevendegree-of-freedom redundant robot with a spherical wrist. This scheme determines the joint velocities required to follow a specified end-effector trajectory while optimizing a given performance criterion using the gradient projection method. The need to determine the generalized inverse of the Jacobian is eliminated and closed-form solutions for joint velocities are obtained. This scheme is well suited for real-time implementation and was implemented on the seven degree-of-freedom CESAR manipulator developed at Oak Ridge National Laboratory. Also presented are graphical simulation results demonstrating the effectiveness of this control scheme for the CESAR manipulator. Extension of this control scheme is currently being addressed for robots with multiple degrees of redundancy. Fig. 6. Joint limit example - least-norm velocity solution. Fig. 7. Joint limit example - performance measure optimized. 35

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