A Dual Stage Planar Cable Robot : Dynamic Modeling and Design of A Robust Controller with Positive Inputs So-Ryeok Oh∗ , Kalyan Mankala∗ , Ph.D. Students Sunil K. Agrawal∗ , Ph.D., Professor James S. Albus∗∗, Senior NIST Fellow ∗
∗∗
Department of Mechanical Engineering University of Delaware. Email: oh,
[email protected]
National Institute of Standards and Technology Gaithersburg, MD 20899-8230 Email:
[email protected] October 4, 2004 Abstract
Cable robots have potential usage for loading and unloading of cargo in shipping industries. A novel sixdegree of freedom two-stage cable robot has been proposed by NIST for skin-to-skin transfer of cargo. In this paper, we look at a planar version of this two-stage cable robot. The disturbance motion from the sea is considered while modeling the dynamics of robot. The problem of robust control of the end-effector in the presence of unknown disturbances, along with maintaining positive tensions in the cables, is tackled using redundancy of cables in the system. Simulation results show the effectiveness of the control strategy.
1
Introduction
Skin to skin transfer of cargo is a challenging task. This transfer operation in mid sea can be unstable unless the sea motions are properly accounted for. Also, one of the basic problems in cable robots is maintaining positive tensions in cables, while manipulating the end-effector. In a typical skin-to-skin transfer operation, the end-effector and the target are located on different ships and hence subjected to disturbances, which may not be in phase. NIST has proposed a two-stage cable robot, each with six-degrees-of-freedom for this purpose. The upper stage is motivated from keeping the cables away from the target containers, and to increase the system redundancy. The lower stage is designed to engage the container. This two-stage design promises to achieve the desired goal of transfer of cargo from one ship to another safely (see Fig. 1). In this paper, we look at the planar version of this robot to develop a better understanding of the scientific problem, which can then be extended to the general problem. During the last few decades, some researchers have studied wire suspended mechanisms. Li et al. [1] presented new models of a continuous backnone robot driven by cables. Hong et al. [2] examined a systematic way of representing complex cable-pulley mechanism configurations and a method to analyze their motion. Sugiyama et al. [3] introduced a nonlinear finite element method for the large deformation and rotation of cables. Nayfeh et al. [4] designed and implemented a controller that suppressed cargo pendulation on most common commercial cranes. Shiang et al. [8] investigated a four cable robotic crane to provide improved cargo handling. Based on the concept of parallel platform, NIST has developed the ROBOCRANE [6] which can control the position and force on heavy machinery, in all six-degrees-of-freedom. In addition, several efforts have been devoted to finding an efficient control strategy and characterizing the workspace with positive cable tensions. Based on the nullspace of the Jacobian, workspace were studied that keep
1
Figure 1: A concept of the proposed crane. cables in tension ([7]-[10]). A force distribution method was proposed to avoid slackness and excessive tension in cables [11]. The dynamic workspace for specific directions of motion and accelerations were studied [12]. The null space of the Jacobian, i.e. the redundancy, satisfies various secondary criteria such as avoidance of joint limits ([13],[14]), avoidance of obstacles ([15]-[17]), kinematic singularities [18], minimization of actuator joint forces [19], or the motion and force control of end-effector ([20],[21]). In this paper, we present a dynamic model for a dual stage planar cable robot, incorporating the disturbance from the sea condition. In addition, a robust controller for effective end-effector control is developed. The problem of positive cable tensions is tackled using redundancy. Since the primary task is to control the end-effector to track the target’s motion, we can treat the orientation of the upper platform as a secondary variable in the system. This extra freedom not only ensures cables to be in tension but also limits the uncontrolled variable using a simple rule. The dynamic model for the dual-stage planar cable robot, incorporating disturbance from the sea condition, is presented in the first section. The next section deals with controller for robust tracking of the desired end effector trajectory in the presence of disturbance. Simulation results are presented to show the effectiveness of the proposed control algorithm.
2
System Dynamics
Fig. 2 shows a schematic of a planar dual-stage cable robot. The two stages, B and C, are connected to frame A by 6 cables. Six actuators are mounted on the frame A to control the tensions in 6 cables. Out of the six cables, three cables are directly connected to the upper stage B. The other three cables pass through three pulleys on the lower stage C and are attached to the stage B. In general, since frame A is attached to a ship, it is not inertially fixed. Due to the sea condition, a ship is subjected to a disturbance motion. This disturbance motion coming on to the frame A has to be modeled while considering the dynamics of the cable structure. It was shown in [22] that if the motion of the end-effector C 2
relative to the coordinate frame A, the disturbance terms characterizing the motion of the upper stage A appear as additive terms in the dynamic equations. This feature is desirable in the design of robust controllers.
Figure 2: A sketch of a planar dual stage cable robot.
2.1
Kinematics
Consider an inertial coordinate frame N with origin ON and basis vectors nx , ny, nz . Body A has a coordinate frame fixed to it with origin OA and basis vectors ax , ay , az . The upper stage has a coordinate frame B fixed to it with origin OB and basis vectors bx, by , bz . Similarly, the end-effector has a coordinate frame C fixed to it with origin OC and basis vectors cx , cy , cz . Note that the origins of the coordinate frames coincide with the center of mass of the bodies. Since the robot is planar, rotation is only permitted about an axis normal to the plane. Note that in this case, nx is the axis normal to the plane. The position, OA of A, is described by xA = (yA , zA )T along the coordinate directions ny , nz . Its orientation is described by angle ψA about nx . So the configuration of frame A, at any instant is described by XA = (yA , zA , ψA)T . For body A, ny a N = RA(ψA ) y , (1) nz az
where N
CψA RA = SψA
−SψA . CψA
Also, we can write the angular velocity of A w.r.t. N as, ωA = ψ˙ A ax = ψ˙ A nx . 3
(2)
We define the position and orientation of B and C in A, by Xi = [xTi , ψi ] = [yi , zi , ψi ]T , i = b, c. Small letters are used to indicate that the position and orientation quantities are described with respect to frame A instead of the inertial frame N . Let A RB and A RC define rotation matrices from frame B to frame A and from frame C to frame A, respectively. In the following analysis, for the purpose of compactness, we will replace, N RA by RA (subscript is capital because it is defined in inertial frame, N ), A RB by Rb , A RC by Rc (subscripts are small because they are defined in frame, A). For example, position of OB , origin of frame B, in frame N can be found as xB = xA + RA xb . The angular velocity of B w.r.t. A is given by ωB/A = ψ˙ b bx = ψ˙ b nx . Simlilary, angular velocity of C w.r.t. A is given by ωC/A = ψ˙ c cx = ψ˙ c nx . Angular acceleration of B, αB in inertial frame, N , can be derived as follows, ωB
=
⇒ αB
= =
ωA + ωB/A d d ωB = ωA + ωB/A dt dt (ψ¨A + ψ¨b )nx
(3)
Similarly, we can derive angular acceleration of C, αC in inertial frame N as αC = (ψ¨A + ψ¨c )nx
(4)
Fig. 3 shows cable attachment points C1 , C2 , C3 on the end-effector, B1 , · · · , B6 on the upper stage B, and A1 , · · · , A6 on body A. Let ai , bi , ci denote the vectors from OA , OB , OC to the cable attachement points, Ai , Bi , Ci repectively, expressed in their local frames, A, B and C. Let us also define three vectors qi connecting B to A, three vectors vi connecting C to A, and three vectors wi connecting C to B (see Fig. 3), in inertial frame, N . Note that these vectors are given by only two components (along y and z directions). −−−→ −−−→ −−−→ q1 = B1 A1 q2 = B2 A2 q3 = B3 A3 − − − → − − − → −−−→ v1 = C1 A4 v2 = C2 A5 v3 = C (5) 3 A6 −−−→ −−−→ −− −→ w1 = C1 B4 w2 = C2 B5 w3 = C3 B6
Figure 3: A sketch of a planar dual stage cable system showing the attachment points. 4
Using loop closure, we can express the vectors qi , vi , wi in terms of cable attachment points and origins of local frames. For example, we can express q1 connecting B1 to A1 , in the inertial frame N as q1 = RA A q1 = RA ¯ q1 , where q ¯1 = (−Rb b1 − xb + a1 ). See Appendix for detailed derivations. Similarly, we can express vi = RA v ¯i and wi = RA w ¯ i , where i=1,2,3. ˆ i , vi = vi v ˆi , and wi = wi w ˆ i , i = 1, 2, 3. So the length li of cable i is given by we can also write, qi = qi q i = 1, · · · , 3. qi (x), li = (6) vi−3 (x) + wi−3 (x), i = 4, · · · , 6. where x = [xA xb xc ]T . On defining l [l1 , · · · , l6 ]T , the position kinematics of the robot is captured in the following nonlinear map, l = l(x) . (7)
2.2
Rigid Body Dynamics
Force Equation Frame A is attached to a ship. Because of the relatively large mass of the ship, the tensions in the cables do not induce any motion in A. The motion of A is only because of the disturbance from the sea condition. Here, we write equations of motion only for the upper stage and the end-effector. By doing force balance on the upper stage, we get the following equations of motion, mb
d2 xB dt2
= =
Fext
mb g + [ q ˆ1
q ˆ2
q3 ˆ
−w ˆ1
−w ˆ2
−w ˆ3 ] T ,
where T is the vector defining tension in different cables whose directions are given by the coefficient matrix. We know, xB = (xA + RAxb ). Also, we can write, q ˆi = qi /qi = RA ¯ qi /qi . Similarly, we have v ˆi = R A v ¯i /vi , w ˆ i = RA w ¯ i /wi . Substituting these in the above equation, we get Dx,b
¨ Axb + 2R˙ Ax˙ b − RT mb g ¨ b + mb RTA x ¨A + R mb x A q¯1 ¯q2 q¯3 w ¯1 w ¯2 w ¯3 = q1 q2 q3 − w1 − w2 − w3 T .
(8)
As already mentioned, frame A is subjected to disturbance from the environment. This disturbance manifests as perturbation in xA and RA . Assuming that we know cable lengths at any time, Dx,b is the only term in the above equation that gets affected because of this disturbance. So this term can be treated as disturbance for the design of robust controllers for T. 2.2.1
Moment Equation
Doing Moment balance about the center of mass of the upper stage (OB ), we get the following equations of motion IB αB = Mext ,
(9)
where IB is the moment of inertia of body B about bx passing through OB and Mext is the moment due to the tension forces acting at the attachment points on the body B. For example, the moment due to tension T1 acting at attachment point B1 is given by (rB1 × q ˆ1 )T1 = (rB1 × q1 ) Tq11 , where rB1 is the position vector from OB to the attachment point B1 expressed in inertial frame. The cross product is defined if the vectors rB1, q1 are defined. Since we have defined the vectors in two dimensinal space (ny , nz) we can interpret the cross product as the vector in the out-of-plane direction, nx with magnitude given by (rB1x q1y − rB1y q1x). This quantity does not change even if the vectors rB1, q1 are expressed in the local frame A. This is because the magnitude of the cross product of two vectors lying in a plane only depends on the relative angle between the vectors. It does not depend on the coordinate frame they are expressed in as long as these coordinate frames are obtained by the rotation about the fixed out-of-plane axis, which is normal to the two vectors considered. So we can write rB1 × q1 = A rb1 × A q1 . But A rb1 = Rb B rb1 and A q1 = q ¯1 . Following the earlier interpretaion of cross-product we can write (rB1 × q1 ) 5
= (Rb B rB1)x q¯1y − (Rb B rB1)y q¯1x nx = CB1 q1 nx(say for notational simplicity). Note that these vectors do not get affected by the disturbance on A. Following a similar approach to obtain the moment due to other tension forces we can write, Mext = q11 CB1 q1 · · · − w13 CB6 w3 T (10) From Eqs. (9) and (10), we get Dψ,b
IB ψ¨b + IB ψ¨A =
1 q1 CB1 q1
· · · − w13 CB6 w3 T
(11)
As in the force equation, even here Dψ,b is the only term that depends on the perturbation in frame A. So we can treat this term as disturbance term to design robust controllers. We can combine both force and moment equations as
Mb
mb I2 01×2
02×1 IB
¨b x ψ¨b
Db
q1 /q1 ¯ Dx,b + = 1 Dψ,b q1 CB1 q1
Jb
· · · −w ¯ 3 /w3 T, · · · − w13 CB6 w3
(12)
where, I2 and 0i×j are a 2 × 2 identity matrix and a i × j zero vector. Similarly, we can write the equations of motion for end-effector as
Mc
mc I2 01×2
02×1 IC
¨c x ψ¨c
Dc
Dx,c v ¯ /v + = 1 1 1 Dψ,c v 1 C C1 v 1
Jc
· · · −w ¯ 3 /w3 T. · · · − w13 CC3 w3
(13)
Note that in the above equation, in the term CCi vi , the subscript Ci goes from C1 to C3 , and then repeats again from C1 to C3 instead of from C4 to C6 . This is because vi and wi cables pass through the same attachement point, i.e., there are only three attachment points on the end-effector. Combining Eq. (12) and Eq. (13) leads to ¨ + D(x, XA ) = J(x)T, Mx M= D(x, XA) =
(14)
Mb O , O Mc Db Jb , J(x) = . Dc Jc
Note that x = [xb , ψb , xc , ψc].
3 3.1
Feedback Controller Sliding Mode Control
The theory of variable structure systems (VSS) with sliding mode has been studied in detail during the last thirty years. It rests on the concept of changing the structure of the controller in order to obtain desired response. There are several advantages in VSS; e.g. high speed response, good transient performance, insensitive to certain parameter variations and external disturbances [24], while standard control schemes such as the computed torque or inverse method is very sensitive to parametric uncertainty, i.e., to imprecision on manipulator inertias, geometry, loads, or friction terms. Hence, the VSS approach has been widely applied to the design of many practical control systems, such as servo system, robot manipulators, and flight control systems, etc. The sliding mode control (SMC), which belongs to a class of VSS, is considered for the control of the dual-stage cable robot in the presence of uncertainties, since SMC not only provides a robust and accurate response, but also makes the system response insensitive to changes in the system parameters and load disturbances. In the following derivation, it is assumed that all the cables keep positive tensions during the motion. Also, the actuators 6
are ideal and cable stiffness is longitudinally large to instantaneously carry the wrench torque to the end-effector. In addition, the control states x and the motion of taget xd are detected by a camera attached to the rotator. First of all, we define a sliding surface and Lyapunov function in the following equations. (15) s6×1 = x˙ − x˙ d + Λ(x − xd ) , 1 T (16) V = s s, 2 λ1 O .. . Differentiating Eqn. (16) w.r.t time leads to where xd is a desired trajectory for x and Λ = . O λ6 V˙
=
sT s˙ ,
=
¨−x ¨ d + Λx ˜˙ ) , sT ( x
=
¨ d + Λx ˜˙ ], sT [ M −1 (J(x)T − D(x, XA ) ) − x
(17)
˜˙ = x˙ − xd . To make V˙ negative, we select the control law as where x where K =
k1
..
O
.
T = J −1 (x) M (¨ xd − Λx ˜˙ − Ksgn(s) ) ] , sgn(s1 ) O .. , sgn(s) = , which leads to . k6 sgn(s6 ) V˙
= ≤
sT [−M −1 D(x, XA ) − Ksgn(s)] , 6 i=1
|si |(
6
|(M −1 )ij |Dj − ki ) ,
j=1 6
−
=
i=1
We select ki =
6
(18)
(19)
ηi |si | .
|(M −1 )ij |Dj + ηi and |D(x, XA)|i ≤ Di , i = 1, · · · , 6. The total energy decreases since V˙ ≤ 0.
j=1
The invariant set satisfying V˙ = 0 has only si = 0 as its candidates. Hence, there does not exist any other points where system may get stuck. Hence, the equilibrium at xd is globally asymptotically stable as long as cables are in tension during motion. To avoid chattering, we replace the sgn() function by the sat() function given as sgn( si ), |si | > Φo si sat( ) = si Φo , (20) Φo Φo , otherwise where we used a constant boundary layer thickness as Φ = Φo [1, · · · , 1]T . Hence the control law now looks like ˜˙ − Ksat( Φso ) ) ] . T = J −1 (x) M (¨ xd − Λx
3.2
(21)
Bound on disturbance
To design successfully the proposed controller for the uncertain system given in Eq. (15), we have to estimate the bound of the disturbance term D(x, XA). The upper limit of D(x, XA) can be calculated as follows ¯ B1 + d ¯ B2 + d ¯ B3 + d ¯B4 ) mb (d IB d¯B5 D(x, XA ) ≤ ¯ ¯ ¯C3 + d ¯ C4 ) , mc (dC1 + dC2 + d IC d¯C5
7
(22)
where sψA y¨A y¨A cψA + z¨A sψA cψA = = −sψA cψA z¨A −¨ yA sψA + z¨A cψA 1 ¯ B1 , ≤ |¨ =d yA |2max + |¨ zA |2max 1
¨A RTAx
sψA ˙ −sψA −cψA y˙ b cψ A ψA = −sψA cψA cψA −sψA z˙b 0 −1 y˙ b |z˙ b | ˙ ˙ ¯B2 , = ψA =d ≤ |ψA |max |y˙ b | z˙b 1 0
(23)
RTA R˙ Ax˙ b
sψA d −sψA cψA = ψ˙ A −sψA cψA dt cψA 0 −1 −1 0 2 ¨ ˙ = ψA + ψA , 1 0 0 −1
¨A RTAR
−cψA −sψA
(24)
zb y 2 ˙ − ψA b −yb zb |zb | |yb | 2 ˙ ¯ B3 , ¨ + ψA max =d ≤ |ψA |max |yb | |zb |
(25)
¨ Axb RTA R
RTA g
= ψ¨A
cψ A = −sψA
sψA cψA
0 1 ¯ B4 , ≤g =d g 1
ψ¨A ≤ |ψ¨A |max = d¯B5 .
(26)
(27)
(28)
¯ Ci , d¯C5 , i = 1, · · · , 4) on the disturbance vector D(x, XA ) have similar forms to The remaining terms (d ¯ dBi , dB5 , i = 1, · · · , 4. In summary, given the minimum and maximum values of the positions and the orientation of the frame A, D(x, XA) can be bounded by a state independent vector D. If we prescribe the motion of the frame A as a sinusoidal function of the form Ai sin(ωi t), i = x, y, ψ, D(x, XA) is bounded by A ω 2 + Az ωz 2 + (Aψ ωψ + Aψ ωψ 2 )|zb | + (Aψ ωψ )2 |yb | + g y y Ay ωy 2 + Az ωz 2 + (Aψ ωψ + Aψ ωψ 2 )|yb | + (Aψ ωψ )2 |zb | + g Aψ ωψ 2 D= Ay ωy 2 + Az ωz 2 + (Aψ ωψ + Aψ ωψ 2 )|zc | + (Aψ ωψ )2 |yc| + g . A ω 2 + A ω 2 + (A ω + A ω 2 )|y | + (A ω )2 |z | + g y y z z ψ ψ ψ ψ c ψ ψ c Aψ ωψ 2 (29)
3.3
Positive cable tension
The designed controller based on sliding mode theory can asymptotically stabilize the system to xd (t) as long as cables are in tension during the motion. A negative tension makes a cable slack, which leads to performance deterioration and even instability. Hence, in this section, we present a method to design the robust controller with positive input. Since the primary task is to control the end-effector to track the target’s motion, we need 8
not control the orientation (ψB ) of the upper platform. This introduces redundancy in the system. This extra freedom can be used to achieve the positive tensions in the cables. Now, the dynamic equation looks like Ms (x)¨ xs + Ds (x, XA ) = Js (x)T .
(30)
Note that xs = [xb , xc , ψc]. The subscript s is used to point out the reduced system dynamic equations which doesn’t include the ψb motion of the upper plaftform. The control law T in Eq. (30) can be written as a sum of a minimum norm solution and an additional feedback gain consisting of the nullspace of input matrix Js (x). T = T + N (Js )m .
(31)
Here, T is the minimum norm solution using the pseudo inverse of matrix Js (x) and is given by ˜˙ s − Ks sat( Φsss ) ) ] , T = JsT (Js JsT )−1 [ Ms (¨ xsd − Λs x
(32)
In Eq. (31), N (Js ) is the null space or kernel of matrix Js (x) and m is a variable to be determined. We will briefly review how to characterize the feasible region of the nullspace variable m to achieve positive tensions [23]. On using the input constraint T(t) ≥ 0 for all time t > 0, the resulting condition is T + N (Js )m ≥ 0.
(33)
Since Js (x) is nonlinear in x, it is hard to get a solution that is globally valid in the state space. However, it is possible to get the solution at a specific point x in the state space. It is clear from Eq. (33) that a feasible solution at a specific x is characterized by a convex region bounded by n linear inequalities on the elements of m. In the following discussions, we assume that the matrix Js has full row rank of 5. With one redundancy, the six linear inequalities in m become n11 u1 u2 n21 u3 n31 m ≥ 0. + u4 n41 u5 n51 u6 n61
(34)
So, the feasible region FA of m is described by the common interval bounded by six linear inequalities. If FA is empty, the tension constraints can not be met.
Minimizing the uncontrolled state Additional feedback gain (N (Js )m) can be also used to minimize the uncontrolled motion of the upper platform, while keeping the cable from being slack during the motion. On using the control law T(t) into the moment equation corresponding to ψb (Eq. (11)) leads to IB ψ¨b + Dψ,b = Jψ,b [T + N (Js )m]
(35)
From the above equation, it can be seen that at any state x, ψ¨b behaves linearly w.r.t m. So, by adjusting m within it’s feasible region we can bound the orientation of the upper platform. Simple rules can be used to select the best acceleration ψ¨b to minimize the uncontrolled motion (ψb ) of frame B at each sample time.
4
Simulations
A simulation for the dual-stage planar cable robot with the sliding mode controller was developed in Matlab Simulink. All the parameters used in the simulations are listed in Table 1. mb and mc are the masses of the upper stage and the end-effector respectively. Ii is the moment of inertia of a stage i, and rij stands for the position vector between j th cable attachment point and the origin on a frame i. In this simulation, we prescribed the unknown motion of the frame A as a simple periodic function xA = 0.05∗sin(0.5∗ 9
Table 1: Simulation parameters In MKS unit. Param. mb mc IB IC ra1 ra2 ra3 ra4 ra5 ra6
Value 100 500 33.67 1150 (2, 0) (1.0, 0) (−2, 0) (2, 4) (−1.0, 4) (−2, 4)
Param. rb1 rb2 rb3 rb4 rb5 rb6 rc1 , rc4 rc2 , rc5 rc3 , rc6
Value (0.6, 0) (0, 0) (−0.6, 0) (0.6, 0) (0, 0) (−0.6, 0) (0.8, 0) (0, 0) (−0.2, 0)
πt)[1, 1, 1]T (see Fig. 8). From the result of the previous section, D = [10.02 + 0.0625|zb| + 0.0025 ∗ |yb |, 10.02 + 0.0625|yb| + 0.0025 ∗ |zb|, 0.025, 10.02 + 0.0625|zc| + 0.0025 ∗ |yc|, 10.02 + 0.0625|yc| + 0.0025 ∗ |zc|, 0.025]T . We choose the set point xd = [0.35 + 0.05 ∗ sin(0.1πt), 0.55 + 0.1 ∗ sin(0.1 ∗ πt), 0, 0.1 ∗ sin(0.4 ∗ πt), 1.6 + 0.1 ∗ sin(0.1 ∗ πt), 0]T and initial states x0 = [0.3, 0.7, 0, 0, 1.5, 0]T . The control gain k to drive the system to follow the desired signals under disturbances from Eq. (19) can be calculated as follows: 1 10.02 + 0.0625|zb| + 0.0025 ∗ |yb | 1 10.02 + 0.0625|yb| + 0.0025 ∗ |zb | 0.025 1 k= + ηo 1 10.02 + 0.0625|zc| + 0.0025 ∗ |yc| 1 10.02 + 0.0625|yc| + 0.0025 ∗ |zc| 1 0.025
(36)
The primary task of the control system is for the end-effector to track the target’s motion specified as sinusoids. In the first simulation, the robust controller is designed to control the six degree-of-freedom motion (x) of the system. In this case, the system is fully actuated by six cables and there is no freedom to handle the positive input constraints. The graphs of desired signals and the tracking performance are shown in Fig. 4. Tension in the first and third cables oscillate between -200N and 200N, -700N and 1.4kN, respectively (Fig. 5), which show that the cables are slack during a part of the trajectory. The second simulation was performed to show the effectiveness of the proposed control strategy, which uses the system redundancy to satisfy the positive tension and minimize the uncontrolled state. As explained in the previous section, the partial state vector xs was used to artificially create the system redundancy. Using the null space’s feasible area (see Fig. 8), we were able to achieve the positive tensions in the cables, as shown in Fig. 7. In addition, by a selection rule, the uncontrolled motion of the upper platform, ψb was bounded between −20o and 20o as shown in Fig. 6. The numerical results lead to the conclusion that the proposed method is useful in designing control systems for achieving the positive tension in the cables.
Conclusions This paper dealt with a novel dual-stage planar cable robot with two moving platforms connected in series. Such cable robots can be used for skin-to-skin transfer of cargo in mid sea. The disturbance motion from the sea is considered while modeling the dynamics of robot. A robust controller was developed which can assure robust tracking of target in the presence of disturbances. Since the primary task is to control the end-effector to track the target’s motion, we neglect the orientation of the upper platform and introduce redundancy into the system. This redundancy is then used to assure positive tensions in cables and keep the uncontrolled state within bounds using rules. Simulation results were presented to show the effectiveness of the controller.
10
Figure 4: State trajectories of the fully actuated system : desired signal(solid), actual signal(dotted)
5
Acknowledgment
The authors appreciate financial supports of NSF award No. IIS-0117733, NIST MEL award No. 60NANB2D0137, and PTI/NIST award No. AGR20020506.
References [1] Changquing Li and Christopher D. Rahn, Design of Continuous Backbone, Cable-Driven Robots, Journal of Mechanical Design, Vol. 124, No. 2, pp. 265-271, 2002. [2] Demmis W. Hong and Raymond J. Cipra, A Method for Representing the Configuration and Analyzing the Motion of Complex Cable-Pulley System, Journal of Mechanical Design, Vol. 125, No. 2, pp. 332341, 2003 [3] Hiroyuki Sugiyama, Aki M. Mikola, and Ahmed A. Shabana, A Non-Incremental Nonlinear Finite Element Solution for Cable Problems, Journal of Mechanical Design, Vol. 125, No. 4, pp. 746756, 2003 [4] Nayfeh, A.H., Masoud, Z.N. A Supersmart Controller for Commercial Cranes, Newsletter, International Association for Structural Control, Vol. 6, No. 2, 4-6, 2002. [5] Shiang, W., Cannon, D., Gorman, J., Dynamic Analysis of the Cable Array Robotic Crane, Proceedings of the IEEE International Conference on Robotics and Automation, Detroit, Michigan, 2495-2500, 1999. [6] Albus, J., Bostelman, R., and Dagalakis, N., The NIST Robocrane, Journal of Research of National Institute of Science and Technology, Vol. 97, No. 3, 373-385, 1992. [7] Williams, R.L.II, and Gallina P, Planar Cable-Direct-Driven Robots, Part I: Kinematics and Statics, Proceedings of the 2001 ASME Design Technical Conferences 27TH Design Automation Conference, DETC2001/DAC21145, 2001. [8] Shiang, W., Cannon, and D., Gorman, J., Dynamic Analysis of the Cable Array Robotic Crane, Proceedings of the IEEE International Conference on Robotics and Automation, Detroit, Michigan, 2495-2500, 1999. [9] Lafourcade, P., Llibre, M., and Reboulet, C., Design of a Parallel Wire-Driven Manipulator for Wind Tunnels, Proceedings of the WORKSHOP on Fundamental Issues and Future Research Direction for Parallel Mechanisms and Manipulators, Quebec, Canada, 1-7, 2002.
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Figure 5: Input trajectories of the fully actuated system [10] Zheng, Y.Q. and Liu, X.W., Workspace Analysis of a Six DOF Wire-Driven Parallel Manipulator, Proceedings of the WORKSHOP on Fundamental Issues and Future Research Direction for Parallel Mechanisms and Manipulators, Quebec, Canada, 287-293, 2002. [11] Shiang, W., Cannon, and D., Gorman, J., Optimal Force Distribution Applied to a Robotic Crane with Flexible Cranes, Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA, 1948-1954, 2000. [12] Barrette, G. and Gosselin, C. M., Kinematic Analysis and Design of Planar Parallel Mechanisms Actuated with Cables, Proceedings of ASME Design Engineering Technical Conference, 391-399, 2000. [13] Fournier, A., Generation de mouvements en robotique-application des inverses generalisees et des pseudo inverse, These d’etat, Mention Science, Univ. des Science et Techniques du Languedoc, Montpellier, France, 1980. [14] Liegois, A., Automation supervisory control of the configuration and behavior of multibody mechanism, IEEE Transactions on Automatic Control, vol. SMC-7, Dec. 1977. [15] Espiau, B., Collision avoidance for redundant robots with proximity sensors, 3rd Int. Symp. Robotics Research, 94-102, 1985. [16] Hanafusa, H., Yoshikawa, T. and Nakamura, Y., Analysis and control of articulated robot with arms with redundancy, Proc. 8th IFAC World Congress, vol. XIV391, 38-83, 1981. [17] Kircanski, M. and Vukobratovic, M., “Trajectory Planning for Redundnat Manipulators in Presence of Obstachles”, 5th CISM-IFToMM Symp. Theroy and Practice of Robots and Manipulators, 43-58, 1984. [18] Luh, J.Y.S. and Gu, Y.L., Industrial Robots with Seven Joints, Proceedings of the IEEE International Conference on Robotics and Automation, 1010-1015, 1985. [19] Hollerbach, J.M. and Suh, K.C., Redundancy Resolution of Manipulators through Torque Optimization, Proceedings of the IEEE International Conference on Robotics and Automation, 1016-1021, 1985. [20] Oussama, K, A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation, IEEE Journal of Robotics and Automation, Vol. RA-3, No. 1, 43-53, 1987. [21] Raibert, M. and Craig, J., Hybrid Position/Force Control of Manipulators, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. RA-3, No. 1, 43-53, 1987. 12
Figure 6: Stage trajectories of the redundantly actuated system : desired signal(solid), actual signal(dotted) [22] Oh, S.R., Mankala, K.K., Agrawal, S.K., Albus, J.S., Dynamic Modeling and Robust Controller Design of a Two-Stage Parallel Cable Robot, Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, Louisiana, April 26 - May 1, 2004. [23] Oh, S.R., Agrawal, S.K., Cable-Suspended Planar Parallel Robots with Redundant Cables: Controllers with Positive Cable Tensions, Proceedings of the IEEE International Conference on Robotics and Automation, Taipei, Taiwan, September 14-19, 2003. [24] Utkin, V.I., Variable Structure Systems with Sliding Mode: A Survey, IEEE Transactions on Automatic Control, 22, 212-222, 1977.
Appendix Using loop closure method we can express the vectors, qi , vi , wi in terms of cable attachment points and origins of local frames. For example, we can express q1 connecting B1 to A1 , in the inertial frame N as q1 −−−→ But, OB B1 −−−−→ OA OB −−−→ OA A1 ⇒ q1
= =
−−−→ −−−−→ −−−→ −OB B1 − OA OB + OA A1 RA Rb by
=
RA xb
= = = =
RA ay −RA Rb by − RA xb + RA ay RA (−Rb by − xb + ay ) q1 RA ¯
where ¯ q1 = (−Rb by − xb + ay ). We can express v1 connecting C1 to A7 in the inertial frame N as v1 −−−→ But, OC C1 −−−−→ OA OC −−−→ OA A4 ⇒ v1
= =
−−−→ −−−−→ −−−→ −OC C1 − OAOC + OAA4 RARc cy
= = =
RAxc RAaz −RA Rccy − RAxc + RA az 13
(37)
Figure 7: Input trajectories of the redundantly actuated system
Figure 8: Feasible area of the redundantly actuated system = =
RA (−Rc cy − xc + az ) ¯1 RA v
(38)
where v ¯1 = (−Rc cy − xc + az ). Simlarly, we can express w1 as, w1 −−−→ But, OC C1 −−−−→ OB OC −−−→ OB B1 ⇒ w1
= =
−−−→ −−−−→ −−−→ −OC C1 − OB OC + OB B1 RA Rccy
=
RA (xc − xb )
= = =
RA Rb by −RA Rccy − RA (xc − xb ) + RARb by RA (−Rccy − xc + xb + Rb by )
=
¯1 RA w
(39)
where w ¯ 1 = (−Rc cy − xc + xb + Rb by ).
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