Biologically Inspired Joint Stiffness Control Shane A.Migliore,Edgar A.Brown,and Stephen P.DeWeerth
Laboratory for Neuroengineering
Georgia Institute of Technology
Atlanta,Georgia30332
migliore@ece.gatech.edu,ebrown@ece.gatech.edu,steve.deweerth@ece.gatech.edu
Abstract—Biological systems are able to perform move-ments in unpredictable environments more elegantly than traditionally engineered robotic systems.A current limitation of robotic systems is their inability to simultaneously and independently control both joint angle and joint stiffness without electromechanical feedback loops,which can reduce system stability.In this paper,we describe the development and physical implementation of a rvo-actuated robotic joint that us antagonistic,ries-elastic actuation with novel nonlinear spring mechanisms.The mechanisms form a real-time mechanical feedback loop that provides the joint with angle and stiffness control through differential and common-mode actuation of the rvos,respectively.This approach to joint co
ntrol emulates the mechanics of antagonistic muscle groups ud by animals,and we experimentally show that it is capable of independently controlling both joint angle and joint stiffness using a simple open-loop control algorithm.
Index Terms—muscle mechanics,antagonistic actuation, variable stiffness,nonlinear spring,field programmable gate array
I.I NTRODUCTION
A NIMALS are capable of autonomously producing a
wide range of stable movements in environments with unpredictable disturbances.They u muscles as energy-efficient,compliant actuators arranged antagonistically about joints to independently control the levels of differen-tial and common-mode actuation.In a biological context, differential actuation is ud to vary joint angle;common-mode actuation(produced by co-contracting antagonistic muscles)decreas passive compliance(increas stiffness) of a joint[1].Dynamic control of joint stiffness is crucial for animals to adapt to changes in task requirements or in environmental conditions.For example,in human walking, a leg’s joints must be stiffer during stance than during swing to allow for the exchange of kinetic and potential energy[2].Additionally,by lowerin
g leg stiffness during the swing pha of walking,the destabilizing effect of a swing leg perturbation can be reduced.
In robotics,joint stiffness can be produced either actively or passively.Active compliance is achieved using nsors and feedback algorithms to adjust joint positions or joint torques as necessary to maintain the desired end-effector compliance with the environment.The primary limitations of active compliance are feedback control constraints and stabilization problems[3],[4].Simple passive compliance can be achieved by inrting an elastic component between This work is partially supported by NSF Grant#0131612to S. DeWeerth and R.Butera.the actuator and the end-effector.In this ca,the need for feedback is removed,but the joint stiffness becomes a constant function of the mechanical elements ud in the robotic device and is therefore not controllable.A more advanced version of this implementation us variable-stiffness elastic elements to dynamically control joint com-pliance.As an example,in[5],[6],a leaf spring was ud as the elastic element,and its elasticity was incread using a slider to limit the amount of spring that was able to move. The resulting function of slider position to spring elasticity was a complex nonlinear function.
Another approach to passive compliance us antago-nistic actuators[7]–[13]with ries-elastic elements[14]–[16].In this approach,it is desirable for joint stiffness to vary as a function of actuator co-
contraction.Linear springs,however,are not sufficient to create variable stiff-ness becau the resulting joint stiffness they produce is independent of co-contraction.
In biology,joint compliance is a function of muscular co-contraction given that an increa in a muscle’s activation increas the number of parallel elastic elements ud by the muscle,which rais the muscle’s stiffness[17],[18]. To emulate this behavior,the elastic elements must have a nonlinear force-length relationship.Several methods have been propod to produce nonlinear elasticity,including the leaf spring mechanism mentioned previously[5],pulley systems with linear springs[10],conical springs[13],and rolamite springs[8].Although robotic systems have been built using some of the mechanisms[10],[11],[19], feedback nsors were still ud to produce appropriate joint stiffness.The need for feedback ems to stem from a common complication of the nonlinear elastic elements—it is difficult to accurately design them to meet a specific force-length relationship using conventional methods[8]. Theoretical analysis has demonstrated that purely quadratic ,F(x)=kx2)in an antagonistic configuration provide a linear relationship between actuator co-contraction and joint stiffness[7],[12].Note that this is similar to biological joints in which co-activation of opposing muscle groups produces stiffness that is both linear and elastic about a t operating point[20].If we assume that quadratic springs could be reliably manufac-tured,the need for
feedback to t joint stiffness would be removed.Unfortunately,the commercial availability of quadratic springs is limited.
In this paper,we describe a novel design for an elas-tic mechanism that can produce specifiable force-length
Proceedings of the 2005 IEEE
International Conference on Robotics and Automation Barcelona, Spain, April 2005
o o
Fig.1.The mechanics of a rotational joint using antagonistic ries-elastic actuation.
relationships,provided the relationships are positive and
continuous.We physically implement the mechanisms
as part of an antagonistically actuated single degree-
of-freedom(DOF)robotic joint,and we experimentally
demonstrate that a simple feedforward algorithm is capable
of independently controlling both joint angle and joint
stiffness simultaneously.
II.J OINT M ECHANICS
Fig.1shows a mechanical schematic of the joint ud in
this rearch.αandβare the agonist and antagonist rvo
angles,andθis the joint angle.As a convention,positive
rotation is clockwi forαand counterclockwi forβand θ.To avoid cable slack,we define the operating region of the joint to beθ∈{−β≤θ≤α}.
Although linear springs are easier to work with and are
readily available,they do not allow the joint stiffness to be
varied.The stiffness that results from their u is
S linear=R2J(k1+k2)(1) where k1and k2are linear spring constants and R J is the radius of the joint pulley.
唐伯虎点秋香对联
Becau we desire stiffness to be linearly related to co-
,c1(α+β)+c2),we have chon to u elastic elements with a quadratic force-length relationship such that
F(x)=a(x−x0)2+b(x−x0)+c(2) where F is the elements’force,x is the elements’s length, x0is the elements’s resting(no-load)length,and a,b,and c are constants.Using the elements and the mechanical configuration from Fig.1,the equation for T app,the torque applied to the joint by the rvos,is
中国国机集团
T app=R J[aR S(α+β)+b][R S(β−α)+2θR J](3) where R S is the radius of the rvo pulleys.The equation for the joint stiffness function,S=dT app,then becomes S=2aR S R2J(α+β)+2bR2J.(4) In this ca,stiffness is linearly dependent on ,the sum ofαandβ),with a constant offt
Fig.2.The design of the nonlinear elastic device.When a stretching force is applied to the device,the rollers are forced along the expanding contour and stretch is applied to a pair of linear springs(one is not visible in background).
proportional to b.Ideally,this offt term would be removed by tting b=0.0N/cm.This would produce springs such that
F(x)=a(x−x0)2+c(5) however non-idealities during machining tend to yield springs with non-zero values of b.
The equilibrium joint angle of the system,θeq,can be found by computing the angle at which the applied joint torque from(3)is equal and opposite that of an external torque,T load,applied clockwi to the joint:
θeq=
R S
2R J
(α−β)−
T load
2R2J[b+aR S(α+β)]
天注定迅雷.(6)
We can solve(4)and(6)so thatαandβbecome functions ofθeq,S,and T load:
α,β=
S−2bR2J
4aR S R2J
±
R J
R S
θeq+
T load
S
.
(7) III.S YSTEM I MPLEMENTATION
A.Nonlinear Springs
Becau the commercial availability of nonlinear springs with specifiable force-length relationships is extremely limited,we developed a device that produces nonlinear elasticity using standard linear springs.In this design (Fig.2),device stretch is applied by moving the roller guide relative to the frame.This forces the ball bearing rollers to move along the frame’s expanding contours and to extend two linear springs(one visible in the foreground and one in the background)as a nonlinear function of device stretch. By altering the contours,the device can be modified to produce a different force-length relationship.
In Fig.2,F reprents the force created by one of the two rollers(and thus equals half of the total output force of the device)and F s reprents the total force the springs apply to the rollers.The independent variable,x r,is the
horizontal distance from the left-most edge of the frame (as oriented in Fig.2)to the rollers’center points,such that for all calculations x r ≥0.x c is the horizontal distance to the point at which the rollers make contact with the frame,and is given by x c =x r +R sin φ.Similarly,the function y r defines the vertical distance from the center of the frame to the rollers’center points,and y c defines the vertical distance from the center of the frame to the rollers’contact points.Using the definitions,the following relationships can be found:
F (x r )=N sin φ(x r )
(8)F s (x r )=N cos φ(x r )
(9)
F (x r )
F s (x r )
=tan φ(x r ) =y c (x c )(10)
y r (x r )=y c (x c )+R cos φ(x r )
.(11)where R is the radius of the rollers and φis the angle
between the rollers’normal force vectors and the vertical axis.F (x )(from (5))is one half of the overall force-length relationship of the device becau force is transmitted equally through both rollers.F s (x r )defines the sum of the forces exerted by both springs:
F s (x r )=4k [y r (x r )−x 0]+F p
(12)
where k is the linear spring constant,x 0is the resting length of the spring,and F p is the sum of the springs’pre-load force.(The factor of four aris becau two springs are being stretched at both ends when the rollers parate.)Using the definitions,the equation defining the device’s contour,y c (x c ),becomes
y c (x c )2
F p
4k
+y c (x c )
y c (x c )+R 1+y c (x c )2
−
x 0
2 −a
8k x c −R
y c (x c ) 1+y c (x c )2 2−c 8k =0.(13)We manufactured two of the devices using a vertical milling machine (one shown in Fig.3)from aluminum to provide a lightweight,minimally deformable structure.Becau the milling machine requires an explicit tool path to be defined,we computed a numerical solution to (13),the equation describing the expanding contour of the device,using Mathematica 4.1(Wolfram Rearch,Inc.,Champaign,IL).We ud the following values for the equation parameters:k =5.5N/cm,x 0=3.81cm,F p =2.62N,R =0.79cm,a =8.27N/cm 2,and c =2.22N.Note that the joint stiffness function can be changed by modifying the definition of F from (2)before computing the numerical solution.
Both manufactured devices demonstrated strong quadratic behavior (r 2=0.9997and r 2=0.9998).The best-fit equations obtained for them were
F 1(x )=(10.97N/cm 2)x 2−(1.
34N/cm )x +1.73N (14)
Fig.3.Mechanical implementation of a novel nonlinear spring device,which was ud to produce a quadratic force-length relationship.
and
F 2(x )=(10.84N/cm 2)x 2−(2.82N/cm )x +1.94N (15)where F 1and F 2are the device forces and x is the device stretch.Although the equations show that manufacturing imperfections produced absolute errors in the quadratic and constant offt terms,the terms have small relative error (less than 11.0%)so overall performance is mini-mally affected.Additionally,linear terms with significant relative error were introduced,but becau the quadratic terms are larger in magnitude than the linear terms for device stretches greater than 2.6mm,the affect on system performance is minimal.The performance of the robotic joint with the springs is detailed in Sec.IV.B.Robotic Joint
有福气的面相
Our implementation of the robotic joint cloly followed the schematic in Fig.1.The rvos ud were capable of producing 1.27Nm of stall torque (Model HS-5945MG,Hitec,Inc.,Poway,CA).Connections to the springs were made using 1.14mm diameter steel cable to prevent any significant contribution to the elasticity of the nonlinear spring.Joint angle was measured using an optical angle encoder with a 0.25o resolution (Model E2,US Digital Corp.,Vancouver,W A).For some experiments,it was necessary to apply an external load torque.This was performed by applying tension to a steel cable wrapped around the joint shaft.Cable tension was measured using a ries force gauge with a 0.2N resolution.The completed device is shown in Fig.4.C.System Control
Although simpler solutions were possible for this appli-cation,we cho to u a field-programmable gate array (FPGA)to provide feedforward control signals to the rvos.A benefit of using FPGAs is that they can produce high-speed,massively parallel computations independent of a computer.We expect that the capabilities will be necessary as this rearch expands.We lected the Xtreme DSP Development Kit (Nallatech Ltd.,Orlando,FL)as our FPGA platform.It features one 2M gate FPGA (XC2V2000-4FG676,Xilinx,Inc.,San Jo,CA),two 14-bit ADC channels,two 14-bit DAC channels,two
Fig.4.The antagonistically actuated single DOF robotic joint with quadratic ries-elastic actuation.
digital I/O bits,and0.5MB of ZBT-SSRAM.The FPGA
was programmed using the System Generator toolbox for
MATLAB(The Mathworks,Inc.,Natick,MA).
The role of the controller can be reduced to receiving
the three input signals(θeq,S,and T load),performing the
algebraic calculations in(7),and producing pul-width
李易峰多大modulated output signals specifyingαandβ.Becau the
development kit only has two ADCs,T load was specified
as a constant parameter for each experiment,whileθeq and S were specified as analog input signals.To simplify the calculations required of the FPGA,(7)was rearranged and
constant parameters were grouped to create three constants, 1, 2,and 3:
α,β=±
1
R J
S
θeq+
T load
+
1
4aR S R2J
2
S−
b
2aR S
3
.(16)
D.Testing Setup
To perform automated testing on the completed system, we ud a dSpace controller board(dSpace,Inc.,Novi, MI)to provide trajectories forθeq and S while recording the actual joint angle in real time.A wiring schematic of the system,including the dSpace controller board and a passive interface circuit board,is shown in Fig.5.
In the next ction,we u the following testing variables extensively:
•θc reprents the commanded(desired)value ofθeq •θm reprents the measured value ofθeq
•S c reprents the commanded(desired)joint stiffness
Fig.5.The electrical connectivity ud for testing the robotic joint,
Fig.6.The accuracy with which the joint was actuated over a range of stiffness values.As stiffness incread,the variability of the actuation decread substantially.
•S m reprents the measured joint stiffness
IV.S YSTEM A NALYSIS
A.Accuracy of Joint Actuation
The accuracy of joint actuation was analyzed by tting θc=0o and recordingθm when S c was t to a ries of values between0.010mNm/deg and0.700mNm/deg.Ten trials were performed for each stiffness value and the joint was randomly perturbed between each trial to allow it to naturally ttle on itsfinal value.The results of this exper-iment are shown in Fig.6.The standard deviation ofθm was2.37o for low stiffness values(S c<0.176mNm/deg) and was reduced to1.20o for higher values.
When stiffness is low,the variability occurs becau the antagonistic forces ud to drive the joint toθc are low enough that frictional forces and bumps on the springs’contours(created during machining)can significantly hin-der the joint from reachingθc.When stiffness increas,the driving forces are able to overcome frictional forces,which reduces trial-to-trial variability.The joint can,however,still consistently missθc by a few degrees becau the contour bumps create local energy minima that consistently attract the rollers to certain positions.A more refined manufac-turing process should significantly reduce the prence and effect of the bumps.
B.Independence of Joint Angle and Stiffness
To verify that joint angle could be actuated indepen-dent of stiffness,θm was recorded for veral values of
Fig.7.Variations in joint stiffness produced by changing the level of co-contraction atθc=0o.Note that as S c increas,the slope of the torque-angle ,S m)also increas.
θc while S c was swept from0.011mNm/deg through 0.807mNm/deg.Ideally,θm andθc would be iden
tical despite variations in stiffness.In the experiment(data not shown),variability inθm was en in the same range as the variability inherent in the system(θm was always within 3.0o ofθc and the standard deviation ofθm was1.25o). To verify stiffness could be actuated independent of joint angle,we measured the static torque that needed to be applied to the joint to produce a ries ofθm values within ±60o ofθc.Fig.7shows the data collected forfive equally spaced joint stiffness values whenθc=0o.
The datafit linear regressions well(the mean correlation coefficient was0.9935),meaning S ,the slope of the torque-angle traces)was linear about the t joint angle. Moreover,as S c incread,S m followed.To demonstrate that this behavior was independent of the t joint angle,θc, this experiment was repeated forθc=±45o.The results from the experiments are shown in Fig.8as a plot of S c versus S m for each of the three values ofθc.The solid trace in thisfigure corresponds to the linear regression through the points;the dashed trace corresponds to the ideal S c vs.S m relationship with identically matched springs.The data pointsfit the ideal trace with a correlation coefficient of0.9980.
C.Compensation for Applied Loads
In the preceding experiments,no external loads were applied,and therefore the T load parameter of the control algorithm,(16),was t to0.0mNm.In this experiment, the effect of a perturbing external torque onθm is measured with and without appropriate ,with and without T load t to the actual external load).
The top three traces in Fig.9show that an external load applied to the joint without algorithmic compensa-tion caudθm to change with stiffness.This occurred becau the joint angle and joint stiffness signals lost their uniqueness.A detailed explanation of this effect is given by Tresilian[21].The bottom three traces show
that the application of appropriate algorithmic compensation caud Fig.8.The relationship between commanded and measured joint stiffness.Five of the points in this plot reprents the slopes of the linear regressions in Fig.7.The remaining ten points come from similar experiments that were performed atθc=±
45o.The dashed trace reprents the ideal joint performance with identically matched springs.
Fig.9.Compensation for an externally applied joint torque.The top three traces show the effect of non-compensated external torques onθm, while the bottom three traces show the results of the sam
e experiment when compensation was applied.When S c>0.28mNm/deg,the joint was within its operating region,and when S c<0.28mNm/deg,one of its actuation cables went slack,producing significant angle error.
the the mean joint angle to be reduced to0.66o with a standard deviation of0.93o,provided the joint was within its operating region
V.C ONCLUSIONS
Using a novel nonlinear elastic mechanism,we have prented an antagonistically actuated robotic joint that has experimentally demonstrated:
蚯蚓的血是什么颜色
•the independence of joint angle and stiffness
•the high correlation between commanded and mea-sured values of joint angle and joint stiffness •the ability of the control scheme to compensate for applied loads
The topic of antagonistic joint control with nonlinear springs has been addresd previously[7]–[13].However, we believe this is thefirst physically constructed system to
u antagonistic quadratic springs to successfully demon-strate open-loop,independent,and simultaneous control of joint angle and joint stiffness.We believe the three cri-teria are important becau they contribute to both system stability and biological similarity.
The broad focus of this rearch is to improve the basic functionality of robotic joints by incorporating stiffness control that emulates animal muscle mechanics so that robots can be designed to more naturally produce a wide variety of animal movements.Emphasis in traditional robot design is often placed on mimicking animal movements rather than on mimicking the underlying mechanics ani-mals u to produce movement.In general,the resulting robots can only successfully perform the specific move-ments for which they were designed.By taking a biologi-cally inspired approach to actuator design,both the quality and variety of robotic movements will be improved. Future improvements to this approach will u nsors to emulate animal proprioceptive feedback and remove the explicit dependence on a priori knowledge of load magnitude to provide adequate compensation.It should be noted that this feedback will not parallel the feedback others have required to produce and maintain a level of stiffness.Rather,the real-time mechanical feedback loop provided by the nonlinear springs will continue to handle environmental perturbations,while the propod nsors will supply the control algorithm with information nec-essary to compensate for longer-term disturbances such as newly applied loads or orientation changes.
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