Differential Flatness of a Class of $n$-DOF Planar Manipulators Driven by 1 or 2 Actuators

Abstract

where 2 m ; Q 2 m2m is non-Hurwitz; (1) 2 m is a causal forcing term described by the unknown LTI system of differential (12).The vector g(t) 2 m is constrained bŷ (k) + p k01 (k01) + 1 1 1 + p 1 _ + p 0 = F k01 g(t) (k01) + 1 1 1 + F1 _ g(t) + F0g(t) (36) where the numbers p0; p1; . . . ; p k01 are the coefficients of the characteristic polynomial (17), and the matrices F k01 ; . . . ; F 1 ; F 0 2 m2mare to be selected. One can uncouple (35) and (36) where the coefficients C 0 ; C 1 ; . . . ; C k01 are selected to provide any desired convergence rate of g(t) ! 0.Matrices F k01 ; . . . ; F1;F0 2 m2m are calculated by equating similar terms on the left and right hand sides of (39) as. . .from whereSubstituting (40) and (41) into (38) we obtain (19) and (20). As soon as the term g(t) converges to zero, the two linear differential equations _ = Q + (1) + g(t); and _ = Q + (1)will have an identical structures and the same forcing term (1) which proves the convergence ! as claimed. The theorem is proven. Int. J. Syst. Sci., vol. 38, no. 11, pp. 871-878, 2007. [11] A. Levant, "Higher order sliding modes, differentiation and output feedback control," Int. J. Control, vol. 26, no. 9, pp. 924-942, 2003. REFERENCES Differential Flatness of a Class of -DOF Planar Manipulators Driven by 1 or 2 ActuatorsJaume Franch, Sunil K. Agrawal, and Vivek Sangwan Abstract-A fully actuated system can execute any joint trajectory. However, if a system is under-actuated, not all joint trajectories are attainable. In recent years, the authors have actively pursued novel designs of underactuated robotic arms which are both controllable and feedback linearizable. These robots can perform point-to-point motions in the state space, but potentially can be designed to work with fewer actuators, hence with lower cost. With this same spirit, the technical note investigates the property of differential flatness for a class of planar under-actuated open-chain robots having a specific inertia distribution, but driven by only one or two actuators. This technical note addresses the following theoretical question: What placement of one or two actuators will make an n-DOF planar robot differentially flat if it is designed so that its center of mass always lies at joint 2? Index Terms-Differential flatness, robot design, under-actuated robots.

Design of a differentially flat 3R planar under-actuated manipulator with a single input at the second joint

“…Through mass redistribution as proposed in [1] [11], the inertia matrix becomes constant. Using torsional springs at the joints, the structure of the compliance matrix is not enough to achieve the property of differential flatness.…”

“…The torsional spring with constant k 123 relates to the displacement of q 1 + q 2 + q 3 . Similar to the model 4 in [1], the center of mass of link 3 is located at the center of the third joint, the center of mass of links 2 and 3 is located at joint 2 using counter balances.…”

“…For such a design, necessary and sufficient conditions for differential flatness property of an n-DOF planar manipulator driven by 1 or 2 actuators were presented in [1]. In this reference, Theorem 1 states that such a dynamic model with only one input is static feedback linearizable if and only if the input is at the first or the last joint.…”

“…In recent literature, an under-actuated open-chain planar manipulator with revolute joints has been shown to be feedback linearizable with specific choices of mass and inertia distribution. It has been shown that the dynamic model of an n-joint manipulator with center of mass located at joint 2 and with only one actuator is static feedback linearizable if and only if the input is at the first or the last joint [1].In our pursuit for novel designs of under-actuated arms, which are both controllable and feedback linearizable, we investigate the feasibility of designing a 3R planar underactuated manipulator which has a different distribution of compliance [1]. This study shows that this new architecture of 3R planar under-actuated manipulator with one input at the second joint can still be designed to be differentially flat.…”

“…In our pursuit for novel designs of under-actuated arms, which are both controllable and feedback linearizable, we investigate the feasibility of designing a 3R planar underactuated manipulator which has a different distribution of compliance [1]. This study shows that this new architecture of 3R planar under-actuated manipulator with one input at the second joint can still be designed to be differentially flat.…”