Review of Actuators with Passive Adjustable Compliance ...

The intrinsic spring stiffness cannot be altered, but the linkor limb stiffness can be selectively varied using an active controllaw, therefore adjusting the virtual stiffness. Figure 2explains the control of a single actuator. The force applied bythe link is given byF ¼ K act (l s a), (4)where l is the length of the actuator, s is the length of the internalball screw that adjusts the equilibrium position of thespring, and a is the free length of the spring. K act is the intrinsicspring stiffness. If a desired behavior is given byF F o ¼ K des (l l o ), (5)about an operating point (F o , l 0 ), the desired actuator positionis given bys des ¼ l a þ (F 0 K des (l l 0 ))=K act : (6)Thus, a position-control scheme achieves the stiffness controlof a single compliant limb. Controlling the position of dcmotors is very simple and differs from the work by Pratt wherethey controlled the torque on the motors and, in turn, controlledthe impedance. The disadvantages of using an approachto control the virtual stiffness are that the performance islimited by the bandwidth of the controller (e.g., during impactthe hardware stiffness will be felt), and it consumes energy toadjust the position of the spring. One overlooked advantage isthat the compliant spring acts to passively change the transmissionratio of the system. For example, if a force compresses thespring to the left and the desired behavior is to move the limbto the right, then as the spring compresses, the motor mustspin faster thus increasing the transmission ratio at the opportunemoment.Antagonistic-Controlled StiffnessThe best-known example of an antagonistic setup is the combinationof biceps and triceps in the human arm. When thebiceps contracts and the triceps relaxes, the arm is flexed. Whenthe triceps contracts and the biceps relaxes, the arm extends.One of the reasons why an antagonistic setup is required is thefact that muscles can only pull and not push. However, morecan be achieved with this setup: when both biceps and tricepscontract, the elbow becomes stiff; when they both relax, theelbow becomes very compliant and the arm hangs freely. Inreality, the muscles in the human arm are controlled in acontinuous way and, thus, the system can cover a whole rangeof positions and compliant behavior. The biologically inspiredconcept of an antagonistic setup is used in a number ofmechanical actuators to obtain adaptable compliance.The Necessity of Nonlinear SpringsThe nonlinearity of the spring is essential to obtain the adaptablecompliance. To explain this, a simple linear antagonisticsetup (Figure 3) is used. The two springs are linear and havethe same spring constant. In Figure 3, x 0A and x 0B are theAdjustable, passive, compliantactuators will rise in importance fortwo reasons, safe human/machineinteraction, and energy savings.controllable positions when zero force is applied by thesprings (the rest length of both springs is assumed zero). Eachposition can be independently controlled requiring twoactuators. The force on the block in the center is the sum ofthe forces of both springs:The stiffness becomesF ¼ k(x x 0A ) þ k(x 0B x)¼ 2kx þ k(x 0A x 0B ): (7)j ¼ dF ¼ 2k: (8)dxThis result is independent of the controllable parametersx 0A and x 0B , and consequently, the compliance is uncontrollableif linear springs are selected.When two springs with a quadratic characteristic are used,the force isx 0AF ¼ k(x x 0A ) 2 þ k(x 0B x) 2¼ 2kx(x 0A x 0B ) þ k(x 2 0B x 2 0A ): (9)x(a)sFigure 3. Demonstration of the necessity of using nonlinearsprings.(b)(c)x 0Ba = Free LengthK actFigure 2. System for one limb. The limb length is measuredby l, whereas the internal motor adjusts the length s. Thecompliance of the limb is adjusted by varying the length s.lSEPTEMBER 2009 IEEE Robotics & Automation Magazine 85