Robotics and Autonomous ELSEVIER
Robotics and AutonomousSystems 25 (1998) 1-18
Systems
A novel biomimetic actuator system R i c h a r d M. K o l a c i n s k i *, R o g e r D. Q u i n n 1 Department of Mechanical and Aerospace Engineering, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7222, USA
Received4 August 1997;receivedin revisedform 20 February 1998
Abstract
The design of a biomimetic actuation system which independently modulates position and net stiffness is presented. The system is obtained by arttagonisticaUy pairing contractile devices capable of modulating their rate of geometric deformation relative to the rate of deformation of a passive elastic storage element in series with the device's input source. A mechanical model is developed and the properties of the device are investigated. The theoretical results developed are then compared with experimental evidence obtained from a simple prototype model of the system. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Series elastic actuator; Modifiablecompliance;Joint compliance;Robotics;Mechanismdesign
1. Introduction
Recent work by many researchers has demonstrated the advantages of using biologically inspired or biomimetic design methodologies for the design of mechanical systems. This has been most prevalent in the design of robotic systems and prosthetic devices. Several researchers have shown that careful design based upon the structures of existing biological organisms can produce devices which outperform systems designed using more traditional methodologies. The driving force behind this is the fact that the locomotive and manipulative capabilities of biological systems far exceed the capabilities of even the most advanced man made systems. This implies that using biologically inspired designs, systems with superior mechanical efficiencies and stability properties can be produced. An important attribute of biological actuation systems is that they appear to regulate their compliance. Two major reasons have been proposed for this. The first of these reasons, noted by Hogan [21], is, if it is assumed that biological systems are somewhat optimized to minimize energy consumption, elastically storing energy ameliorates the energy costs inherent in many musculoskeletal motions. The second explanation is that it appears to act as a stabilizing mechanism for countering the effects of transmission delays in neurally mediated responses. In support of the first of these propositions, Cavagna and Kaneko [6] found that the metabolic efficiency of a human experimental subject was about 50% whereas the efficiency of muscle is commonly assumed to be no higher than 30%. Cavagna and coworkers [14] found that both humans sprinting and kangaroos hopping can have metabolic efficiencies higher than 80%. Elastic energy storage is credited with the difference in efficiencies. Based * Corresponding author. E-mail:
[email protected] 1E-mail:
[email protected] 0921-8890/98/5- see front matter © 1998 ElsevierScienceB.V. All rights reserved. PII: S0921-8890(98)00034-7
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
upon similar experimental results, Alexander [2] asserts that humans store energy in their Achilles tendons and the ligaments that support the arch of the foot, and that compliant legs and feet reduce the peak forces that occur when the foot strikes the ground. In a similar vein, Cavagna et al. [7] suggest that running is essentially bouncing. Other research suggests that in addition to the storage of elastic energy in the ligaments and tendons, energy is also stored elastically in the muscles themselves. In addition to the results stated above, Cavagna [5] also found that the reflex stiffness of muscle behaves as a lightly damped spring. Houk [22] found that the neural motor servo regulates the muscle stiffness rather than the force or muscle length separately. Hoffer and Andreassen [15,16] found that reflex effects combine with the intrinsic properties of muscle to produce a stiffness that increases rapidly at low levels of applied force. Hogan [21] asserts that muscle behavior should be modeled with a dynamic quantity, mechanical impedance or admittance. He suggests that neural feedback regulates quantities such as position and the apparent behavior of the muscles in order to transform the apparent mass, damping and stiffness of the limbs and joints to compensate for the apparent link geometries as the joint angles vary. The second proposition is supported by the works of Akazawa and Milner [1] and Humphrey and Reed [23] who suggest that the joint stiffness is a mechanism for countering the destabilizing effects of transmission delays in the neural feedback control. Grillner [13] postulated that the ability to control the dynamics at high frequencies stems from the intrinsic muscle impedance itself. He demonstrated that the intrinsic stiffness of the muscle is responsible for the earliest response of the lower limb to perturbations encountered in normal walking. Neural feedback did occur, but later than the stiffness response. These attributes of compliant actuator systems have inspired several researchers to incorporate compliance into their designs. An excellent example of this can be seen in the work of McGeer [27] who constructed a series of what he termed "Passive Dynamic Walking" machines. They can walk down inclined planes, using work from gravity to replace energy losses in dynamically stable gaits. These mechanisms are motivated by toy animals which can walk downhill by waddling from side to side and by the work of Mochon and McMahon [28,29] who were in turn motivated by Cavagna's [5] discovery that the total mechanical energy of the body changes little during a walking step. Espenschied et al. [11,12] designed a hexapod walking machine actuated by DC servo motors and used active, variable joint stiffnesses to permit it to conform to irregular terrain and exhibit reflexive behaviors. Raibert has incorporated series elastic elements in the construction of walking machines [4,8,33] which are designed to mimic the dynamics of walking organisms and can be regulated using simple controllers. Hogan [ 17-20] has developed a compliant motion control technique called "Impedance Control" based upon the supposition that muscle compliance is an integral part of stabilizing the motion of biological systems and their interactions with their environments. Using this technique, Colgate and Hogan [9,10] have shown that the necessary and sufficient condition for a system such as a robot or a human limb to avoid instability when coupled to any passive, stable system is that the driving point impedance at the point of coupling must appear to be that of a passive system. Pratt [31,32] has constructed and implemented a system with an actively variable spring in series with the position (or force) source. The system produces low error below the natural frequency of the system and within this range and increases the shock tolerance of the system, reduces the effects of reflected inertia and stores energy elastically. In a similar approach, Morrell [30] used a pair of coupled actuators in series with elastic elements to extend the dynamic range of actuation in an arrangement analogous to hi-fi speakers. In another similar approach, Sugano and coworkers [34] studied an actuation system where a nonlinear spring element is used as the series element. The literature cited here strongly suggests that the use of flexible elements in series and in parallel with position sources can greatly increase the efficiency of mechanical manipulators as well as serve as a stabilizing mechanism. They can be used to recoup energy that would ordinarily be lost and the stiffnesses will act as a proportional feedback controller whose dynamics are instantaneous and hence can compensate for controller lags and possibly for some of the difficulties inherent in non-colocated control. In addition, it should be noted that an elastic element can also store energy input by external disturbances. An appropriately designed controller would be able to use these perturbations to help drive the system, further reducing the energy requirements of the system.
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
2. Mechanical design The actuator system detailed here is able to store energy and independently modulate force and compliance. Energy storage is accomplished by placing a passive storage element in series with the input source. Feedback control can be used to regulate the apparent stiffness of the system but cannot store energy and in fact may act as a destabilizing mechanism. It is therefore desirable that the modifiable compliance be an intrinsic system property. The inspiration for such a system comes from the antagonistic coupling of muscle in biological organisms. 2.1. Antagonistic actuation Grossly simplifying muscle behavior, a muscle pair can be thought of as two springs, each possessing stiffnesses and unstretched lengths, Ki and Li, respectively, connected to ground and to each other such that they are deformed by lengths 8i and span a distance L0 # Ll + L2, Fig. l(a). Applying compatibility conditions, the equilibrium position in terms of K,~, Li and L0 is =
KIL 1 + K2 (L0 - L2) K1 + K2
(1)
Using the definition of the equivalent stiffness for springs in parallel, and Eq. (1), the stiffnesses K1 and K2 required to produce a desired combination of net stiffness, Kn*t, and equilibrium position, J2*, can be determined as ~* + L2 - Lo K*et, L1 + L2 - L0 L1 -,~* K2 -K*et. L1 + L2 - L0 K1 --
(2) (3)
Note that changing the equilibrium point is dynamically equivalent to applying a force to the system and that while the force applied at the node between the two springs may be regulated by modulating the unstretched lengths of the springs, the net stiffness cannot be. 2.2. Stiffness modulation For a system to have.,a modifiable passive stiffness, it must possess a spring whose mechanical properties or whose rate of deformation relative to the system's deformation is modifiable. Materials whose mechanical properties are modifiable suffer from severe limitations on response speed, range of motion or magnitude of force produced. In the absence of a suitable material, a mechanism which changes the kinematic behavior of the ends of the springs is necessary. The scissor-like mechanism, Fig. 2(a), does exactly this. The spring element is deformed by the load induced motion of the cars to which the spring is attached. Applying a tensile load, Fw, to the cable tendon attached to the spring cars will cause the cars to spread further apart as they move along the scissor links of the mechanism, which move about a common revolute joint at O. Varying the angle these links make with the horizontal, varies the rate of deformation of the spring relative to the rate of deformation of the mechanism, as seen from the applied load. The mechanism and cable tendons through which the force is applied will only support tension; therefore, this device must always be preloaded. Connecting two of these mechanisms together antagonistically will generate the necessary preload and produce a system with modifiable net stiffness and equilibrium position. A smaller effect due to the geometric distortion of the tendon configuration also occurs. It can be seen by inspection that deformation of the:system in the direction of the tensile load will produce a stiffening effect while motion opposite the applied load will produce the converse effect. This is a higher order effect and is negligible compared to the effect of changing the link angles, provided the length of the tendons is sufficient. The existence of a controller sufficiently robust to regulate this system is assumed. The design and construction of such a controller is beyond the scope of this paper, however, the design, implementation and performance of several controllers can be found in [24-26].
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
(a)
Y
EX
L--F Ka6a
K1( ~ 1 ~ L--- F (b)
Fig. 1. (a) Deformedsprings at equilibrium. Co)Schematicof mass/spring system. A schematic representation of a link and a spring car is shown in Fig. 2(b). Assuming a massless spring car and applying Newton's Second Law to the spring car in the direction along the link, the Lt direction, yields !
•
F~,smfl - Fssin0 = 0,
(4)
where F~ is the force of the cable tendon acting on the spring car, Fs is the spring force acting on the spring car, 0 is the angle between the scissor link and the horizontal and fl is the angle between the line of action of F~, and the perpendicular to the scissor link. Next, we define the angle/z = 0 + fl as shown in Fig. 2(b). Applying compatibility conditions at the connection between the tendons to the spring cars and applied load yields Fw = 2Ft~ sin #,
(5)
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
5
(a)
(b) Fig. 2. (a) Stiffnessmodulationmechanism. (b) Springcar detail schematic. where Fw is the tensile load applied to the mechanism. Rearranging Eq. (5) and substituting it into Eq. (4) produces the following relationship between the applied load and the spring force: Fw sin/~ 2 sin/z
Fs sin 0 = 0.
(6)
Next, the spring force is defined in terms of the spring stiffness and deformation, substituted into Eq. (6) and equated to the effective stiffness of the mechanism. This produces, mutatis mutandis: Fw = 4K sin/.t sin 0 (h - h0) = Keff(x - x0), sin B
(7)
where the subscript 0 denotes unstretched lengths and K is the spring rate of the mechanism's elastic element. Fig. 3(a) shows a schematic of the mechanism's geometry above its line of symmetry. Lt is the displacement of the spring car along the scissor link, )~1 is the projection of Lt onto the horizontal, h is the perpendicular distance of the spring car from the horizontal, Lw is the length of the cable tendon connecting the spring car to the loaded
6
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
h
\\
X
(a)
(b) Fig. 3. Schematicsof (a) mechanism; (b) cable tendon geometry.
tendon, )~2 is the projection of Lw onto the horizontal and Y is the total deformation of the mechanism. From the geometry, the following relationships can be seen: X = ) ~ I "~ ~-2,
~1 = h cot 0
=
Lt c o s 0 ,
)~2 = h tan/z.
(8) (9) (10)
Inserting Eqs. (9) and (10) into Eq. (8) produces the following definition of the total deformation of the system: = h c o t 0 + h tan/z,
(11)
R.M. Kolacinski,R.D. Quinn/Robotics andAutonomousSystems25 (1998)1-18 and the effective unstretched length of the system is found by substituting h0 for h into Eq. (11): 20 = h0 cot 0 + h0 tan/x0,
(12)
where tan/z0 = ~/(Lw / h0) 2 - 1 is a design constant reflecting the preload on the system. Note that the effective unstretched length changes with the system configuration as does the effective mechanism stiffness. The mechanism's deformation may now be defined: x - xo = (h - ho) cot 0 + h tan/z - ho tan/zo.
(13)
Defining 32 = x - xo and 3h = h - ho, Eq. (13) can be written as 82 = 8h cot 0 + (h0 + 8h) tan/z - h0 tan/z 0.
(14)
The length 3h is a measure of the geometric distortion of the cable tendon geometry. As stated earlier, the effect of this distortion on t]he mechanism deformation, 3x, is negligible if the cable tendon lengths are great enough. Linearizing Eq. (14) about the point 3h = 0 produces the approximation 32 = (cot 0 - cet/zo)3h + O(3h2).
(15)
Substituting Eq. (15) into Eq. (7) and simplifying produces a definition of the mechanism's effective stiffness: 4 K sin/z sin 0 ( sin/zo ) sin 0 Keff = sin/~(cot 0 - cot/zo) = 4 K \ si--~o ] cot 0 c o t / z '
(16)
where/3o = / z o - 0o, 0o is the nominal scissor link angle. From Fig. 2(b), the distance Lt can be expressed as / _
Lt = 2 cos 0 - ~/L~ - ~2 sin 2 0.
(17)
Substituting Eq. (17) :into Eq. (9) produces an expression for h in terms of 2; h = ~ cos 0 sin 0 - ~/L 2 - ~2 sin 2 0 sin 0.
(18)
Substituting Eq. (18):into the geometric definition of cot ~ yields the following expression:
cot/z=--
cos 0 - J L 2, - ~2 sin 2 0 / "_ 2 sin 0 + ~/LZw _ ~2 sin 2 0 cot 0
(19)
Introducing Eq. (19) into Eq. (16) and simplifying produces the following expression for the mechanism's effective stiffness in terms of its scissor link angle, 0, and deformation, 2: Keff = 4 K ( s i n / z o ~ xsin4 0 q- V/L2 - x2 sin2 0 sin2 0 cos0 \ sin/~0 /
v/L 2 _ ~2 sin 2 0
(20)
It can be seen by inspection that increasing the mechanism's deformation will increase the stiffness as will increasing 0. The stiffness is dominated by the trigonometric terms, however, and the effect of geometric distortion is only significant at the extremes of the mechanism's range of motion. This is partially reflexive of the assumption that the mechanism's deformation is linearly related to the spring deformation. It does, however, accurately reflect the fact that the cable: tendon geometry's deformation has a small effect on the mechanism's total deformation. It
8
R.M. Kolacinski, R.D. Quinn/Robotics andAutonomous Systems 25 (1998) 1-18
Fig. 4. Biomimetic actuation system.
should also be noted here that the range of 0 is limited. The maximum possible range is 0 6 (0, :r/2), though any physical realization of this device will have a smaller maximum possible range. The most important of this mechanism's attributes is that it permits the effective stiffness of the device to be modulated by an external input. Specifically, by modulating the angle 0, the effective stiffness is modulated. As noted in the prequel, the mechanism requires a preload. This is supplied by coupling two of these mechanisms antagonistically, as shown in Fig. 4. The net stiffness of this system can be obtained by noting that the two mechanisms operate in parallel and, therefore the net system stiffness is the sum of the effective stiffnesses of the two mechanisms which comprise it: (sintzol ] x1 sin401 q- x / L 2 1 - x ~ sin2 01 sin2 01 cOs01 Knet : 4KI \ sin to1 .] ( sin/z02 ~ + 4 K 2 \ sin to2 ]
~/L21 - 22 sin E 01 X2
sin'* 02 + ~ L 22 - 22 sin 2 02 sin 2 02 cos 02 (21)
~L22 - 22 sin g 02
The variables 2i are the local deformations of the two mechanisms. If it is assumed that the devices are identical and have the same nominal input angle, then the nominal operating point is the midpoint of the range [0,L0]. Defining a variable x with its origin at this midpoint and applying the continuity condition, (21) can be written as (sin/z0~ I (_Lo/2 + x) sin401 + Knet = 4 K \ sin/3o } L~/L 2 _ (Lo/2 + x ) 2 sin2 01 sin2 01 (L0/2 - x) sin 4 02 1 x cosO1 + ~/L2 _ (Lo/2 - x) 2 sin E 02 + sinE 02 COS02
.
(22)
2.3. System dynamics
Similar to the modeling of the net system stiffness, the dynamic behavior of the system may be modeled as a mass connected to ground via two springs in parallel as shown in Fig. l(b). The equation of motion for this system is mY = F -- Fwl + Fw2
(23)
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
where x is the position of the mass with origin at L0/2, F is an externally applied load and Fw~ and Fw2 are the reaction forces from the left and right elements, respectively. Referring to Eq. (7) and using the geometric identities: sin/z
=
h2
v/~w -
Lw
and
cos/z
=
h
Lw
and the trigonometric identity sin fl = sin tt cos 0 - cos/,t sin 0. The cable tendon load can be restated as
4K (hi ho) -
Fwi
cot Oi - cot ~i
i --- 1, 2.
(24)
Again using the geometric and trigonometric relationships shown in Fig. 2(b), these tensile forces can be expressed in terms of Xi and Oi:
[(-~icosOi -~/LZ --~ZsinEOi) sinZOi -hosinOi] (25)
X
~/L2w _ ~2 _ ~2 sin20i Substituting this expression into Eq. (23) and supplying the local definitions of Xi in terms of x yields the following equation of motion:
m=4lELo,2-x,sin2o Lo,2-x,2sin2O2cosO21 -L+ XF
(L0/2 - x) sin 02 COS 02
L
-
sin 02
--
h01 sin 02
- ( L o / 2 + x)2 sin 2 01 - sin 01 - ho
sin 01
+ F.
(26)
Note that, unlike the derivation of the net system stiffness, these equations of motion do not assume a linear effect from the distortion of the cable tendon geometry.
3. Parametric analysis o f
actuator
system
It is apparent from the complexity of Eq. (22) that back substitution into Eq. (1) is not a viable means of determining the equilibrium point of the system in terms of the mechanism configuration. In order to simplify, the higher order terms due to the geometric distortion of the mechanism are linearized. Solving the resulting equations for the equilibrium position produces: = [ L o / 2 - (cot 02 + tan /z0)](cot 01 -cot/z0) 2 _ [ L 0 / 2 - (cot01 + tan/z0)](cot02 - cot/zo) 2 ( c o t 01 - cot/x0) 2 + (cot 02 - c o t / , t o ) 2 (cot 01 - c o t / , t o ) 2 + (cot 01 - - cot/z0) 2
(27)
R.M. Kolacinski, R.D. Quinn/Robotics and Autonomous Systems 25 (1998) 1-18
10
Fig. 5(a) shows the equilibrium position ~ as a function of the scissor angles 01 and 0e. 01 and 02 vary over the range 0.2rad < Oi < 0.8rad for i = 1, 2. The mechanism's stiffness at these equilibrium positions can then be obtained directly from Eq. (22). Fig. 5(b) displays the stiffnesses corresponding to the equilibrium positions. These plots lead to two important conclusions. The first is that the equilibrium position is axisymmetric about the plane 01 = 02 whereas the stiffness plot is symmetric about that plane. This indicates that the equilibrium position and net stiffness of the system can be independently specified. The second conclusion is that the stiffness becomes undefined for larger values of 01 and 0e, i.e. the net stiffness approaches infinity for larger pairs of input angles. This singularity is reflexive of the geometric constraints induced by the cable tendon lengths. This occurs when either of the radicals in the denominator of Eq. (22) go to zero. This can also be seen in the system's equaions of motion, Eq. (26). This is due to a poor transmission angle in the cable tendons at these extremal input angles. The set of input angles, Oi and position, x for which the stiffness (and E.O.M.) are defined is a compact set, hence a local controller can be defined for this system. The poor transmission angle is achieved for a scissor mechanism when the denominator of Eq. (20) goes to zero. Setting the denominator equal to 0 and solving for 0 produces an expression for the maximum permissible angle: 0max = sin -1
L0/-2--+x
28)
'
Fig. 6(a) shows the maximum scissor angle as a function of element deformation. Note that only positive deformations are examined. By symmetry, negative displacements will cause the same effect in the opposing mechanism when the elements are antagonistically coupled. This observation leads to the conclusion that a given scissor angle in one mechanism can constrain the maximum scissor angle of the antagonist mechanism. For a given agonist scissor angle 01, the maximum possible scissor angle of the antagonist mechanism, 0era,x, is given by
(
LwsinO,
0emax = sin-1 \ L 0 sin0j - LwJ"
(29)
This relationship is shown graphically in Fig. 6(b). The plot is symmetric as expected and shows that small agonist input angles allow large antagonist input angles and large agonist input angles will only allow small antagonist input angles.
4. Stability a n d c o n t r o l l a b i l i t y
The state space representation of the equations of motion, (26), is x2
/i Loj2:Xl,Sin3u2 +sinu2cosu21 L~L 2 - ( L o / 2 - xl)2 sin 2 u2
x [(Lo/2 - x]) sin u2 cos u2 - ho
- . o/2-x,) ingu si.u2]
= -
F(-L--°/2+Xl)sin3ul--sin2
L¢L2w - (Lo/2 + xl) 2
+ sin u 1 cos u 11
Ul
x [(L0/2 + xl) sin Ul cos u2 -- h0
-~/LZw - (Lo/2+xl)Zsin2ul s i n u ! - - h 0 ] / + 8
(30)
R.M. Kolacinski, R.D. Quinn /Robotics and Autonomous Systems 25 (1998) 1-18
ActuatorEquilibriumPosition
60~
"
40~ ~' 20~
g
-20 -40~ -60,
Antagonist Angle, 02 (cad)
0.2
0.2 Agonist Angle,
01(tad)
(a) Actuator Equilibrium Stiffness
30. 25~
~10. m 5.
i
O.
Antagonist Angle, 02 (rad)
0.2
0.2
Agonist Angle, $1 (rad)
(h)
Fig. 5. (a) Actuator equilibriumposition. (b) Actuator equilibriumstiffness.
11
12
R.M. Kolacinski, R.D. Quinn /Robotics and Autonomous Systems 25 (]998) 1-18 Maximum ComplementaP/Scissor Angles 1.6
1.4
~. 1,2 _="
0.8
\
\
0.6
0.4 0.4
r
0.6
i 08
i 1 AgonistAngle,01 (rad)
i 1.2
i
1.4
1.6
(a) MaximumScissorAnglevs Displacement 0.64l
0.62f ' ~ 0.6 ~0.58 0.56
}
-- 0.54 0.52 0.5
0"460
10
2
0
30 ~ 410 MechanismDeformaUon(ram)
i
50
60
(b) Fig. 6. Maximum antagonist input angle versus: (a) agonist input angle; (b) Mechanism deformation.
R.M. Kolacinski, R.D. Quinn / Robotic s and Autonomous Systems 25 (1998) 1-18
13
Table 1 Analysis parameters LO (mm)
Lw (mm)
ho (mm)
K (N/mm)
m (kg)
K(N/kg ram)
/x0 (rad)
3o (rad)
co
510
150
67.32
1.0
0.038293
104.4575
1.1054
0.5818
1.6262
The regulated outputs are system position and net stiffness given by: x1 I
4Kco
Lo/2 + X l ) sin4 u 1
q_ sin2 Ul cos Ul
/
~/L 2 -- (Lo/2
y =
-
x1) 2
sin 2 Ul
(Lo/2 - Xl) sin g u2
+
(31)
d- sin 2 Ul cos u2
F
_ J L 2, - (Lo/2 - Xl) 2 sin 2 u2 B
where the parameter values are obtained from Table 1, with the derived parameters defined; K = 4 K / m , co = sin/zo/sin/~0 and ~ = F / m and u = [0102]T. Differentiating Eqs. (30) and (31) with respect to x and u and evaluating at x = u = 0 yields the following linearization matrices: A= C=
[
0 -150.426
1 0 '
0
D=
01 '
J
B=
[
0 - 2 9 , 415
[0 0] 8.531
8.531
0 29, 415
]'
(32)
"
(33)
The eigenvalues of A, )~1,2 = +12.265j, lie on the imaginary axis and no information on the stability of the system can be inferred. Inspection of the Jacobian shows that the linearization of the system for any input angles and their corresponding equilibrium point will also produce a center and therefore no possible equilibrium point can be shown to be stable. Physically, this supports the intuition that this system can be considered an undamped oscillator. The controllability and observability matrices for this system are
0
--2.9415
0
2.9415
0
2941,] 0 x 104,
(A, C) =
001
il
"
(34)
0 It can be seen that both (A, B) and (A, C) have rank 2 and that therefore, the system is locally controllable and observable. This demonstrates that, at least locally, the position and net stiffness of this system can be independently regulated.
5. Experimental results In order to ascertain the validity of the system model developed here, a simple prototype of the mechanism was constructed. The mechanism, shown in Fig. 7, is manually operated, it has no active elements. The angle the scissor link makes with the horizontal is controlled by manually adjusting the throw of turnbuckles between the ends of the scissor links distal from their common revolute joint. In a fully active version, these turnbuckles would be replaced by a linear actuator such as a hydraulic piston.
14
R.M. Kolacinski, R.D. Quinn /Robotics and Autonomous Systems 25 (1998) 1-18
Fig. 7. Prototype of actuator system. Using the prototype, the validity of the models used to compute the equilibrium position and corresponding stiffnesses was verified. The equilibrium position and stiffness were measured for a set of input parameters over the entire range of inputs admissible by this realization of the actuator. Both 01 and 02 were varied over the range Oi [0.26 0.36] tad, i = 1, 2. The physical parameters of the prototype, given in Table 2, were then used to generate theoretical predictions of the equilibrium position and stiffness over this range. Table 2 Prototype parameters L0 (ram)
Lw (ram)
/z0 (rad)
00 (rad)
479
152.4
1.267
0.329
.R,M, Kolacin#ki, R,D, Quinn/ Robotics and Autonomous Systems 25 (1998) 1-18
15
Expeflmentalvs. Theoretical Eq. Pos. • ..
! •" "
'"
•
" ......... i ' " " ' " . . :
~
....
•
....+
..
: ..........
150 ~
... ,
iI
'
.
100
................. !i . .... .
o E
.
.
.
.
.....................
C
-1~
..........
"........
"
.
...............
0.4 ~
:
"
"
':
0,3 ivttaoonist gn~e, 82 (tad)
:
".:.
" :~"'"
~:
0.45
o3 0.25
0.25
Agonist Angle, 01 (tad)
(a) Experlrnenttlvs, TheoreticalStiffness
0.2;:
0.2 0.1E~
! 0,t~ 0.14. ~ o.1,~! 0.1
0.08
0.06
0.45
0.45
AntagonistAngle, 02 (lad)
0,25
0.25
Agonist Angle, 01 ifad)
(b) Fig. 8. Expel~mental results versus theoretical predictions: (a) equilibrium position; (b) equilibrium stiffness.
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R.M. Kolacinski, R.D. Quinn /Robotics and Autonomous Systems 25 (1998) 1-18
Fig. 8(a) shows the equilibrium position as a function of the input angles. As can be seen, the agreement is excellent over all. At the larger angles, the experimental results differ slightly from the theoretical predictions. It is felt that this is a product of materials and construction of the prototype rather than a shortcoming of the model. The prototype as it currently exists has rather loose tolerances and a great deal of friction which has a pronounced effect on the equilibrium position. In addition, the system's preload is relatively small. This affects the behavior of the system the most when one input angle is near the max of its range while its antagonist is near the min of its. This is manifested in the results where the magnitudes of the experimental results are larger than those predicted by the model. Fig. 8(b) shows the mechanism stiffness corresponding to the equilibrium positions. Again, the experimental results agree well with the model. The model slightly under predicts the stiffness. It is believed that this is the result of the excessive friction in the system and the lack of resolution in the spring scale used to measure the mechanism stiffness. Due to the excessive friction in the system, a more accurate method for measuring the stiffness, such as via measuring mechanical mobility are not viable. Despite the difficulty in measuring the stiffness, the model shows excellent agreement. It should be noted that the prototype mechanism has a very limited range of motion due to the manner of its construction and hence its behavior could not be examined very far from its nominal operating point. In other words, this model was only examined in the region where the linearizing assumptions would be fairly accurate and therefore nothing can be said about the range over which this model is valid. It can, however, be said that the model is valid in a neighborhood of the system's nominal operating point.
6. Discussion and conclusions
The analysis of this actuation system demonstrates that it is capable of controlling the system's equilibrium position (or equivalently the net force produced on the system's central mass) independent of the system's net stiffness. Used in conjunction with a feedback controller, it is possible to specify not only a position (or force) trajectory but a net stiffness trajectory as well. Experimental comparison of the static properties of the system are in agreement with theoretical predictions. Small discrepancies between the experimental results and the theoretical model are readily explained by deficits in the current realization of the system. A more careful design and construction of the mechanism can remove many of these deficits. For instance, the use of linear bearings would decrease the amount of friction in the mechanism and a higher preload would remove some of the larger than predicted displacements that occur when one input is at its maximum and the other input is at its minimum. The mechanism presented has the potential to improve the efficiency of many controlled mechanical systems, particularly those undergoing repetitive motions (i.e. walking machines and manipulator end effectors). By allowing the actuator to control the flow of previously stored mechanical energy, the power an actuator must supply to such a system is reduced to merely that required to replace energy lost through dissipative mechanisms such as friction. This actuator system can be used with any linear actuator such as pneumatic or hydraulic pistons or electric motors. The stability and efficiency characteristics an actuation system such as this possesses has many potential applications, particularly those in which weight considerations drive efficiency needs such as prosthetics and autonomous robots. References [1] K. Akazawa, T.E. Milner, Modulation of the stretch reflex in human finger muscle, in: Proceedings of the Conference of Vocal Fold Physiology, 1981. [2] R. McN. Alexander, Elastic Mechanisms in Animal Movement, Cambridge University Press, New York, 1988. [3] E.K. Antonsson, R.W. Mann, The frequency content of gait, Journal of Biomechanics 18 (1985) 39-49. [4] H.B. Brown Jr., M.H. Raibert, M. Chepponis, Experiments in balance with a 3d one-legged hopping machine, The International Journal of Robotics Research 3 (1984) 75-92. [5] G.A. Cavagna, Elastic bounce of the body, Journal of Applied Physiology 29 (1970) 279-282.
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[6] G.A. Cavagna, M. Kaneko, Mechanical work and efficiency in level walking and running, Journal of Physiology 268 (1977) 467-481. [7] G.A. Cavagna, H. Thys, A. Zamboni, The sources of external work in level walking and running, Journal of Physiology 262 (1976) 639-657. [8] M. Chepponis, M.H. F~aibert,H.B. Brown Jr., Runningon four legs as though they were one, IEEE Journal of Robotics and Automation 2 (2) (1986) 70-82. [9] J.E. Colgate, N. Hogan, Robust control of manipulator interactive behavior, in: K. Youcef-Tourni, R. Shoureshi, H. Kazerooni (Eds.), Modeling and Control of Robotic manipulators and Manufacturing Processes, ASME, 1987, pp. 149-159. [10] J.E. Colgate, N. Hog~aa,Robust control of dynamically interacting systems, The International Journal of Control 48 (1988) 65-88. [11] K.S. Espenschied, R.D. Beer, R.D. Quinn, H.J. Chiel, Biologically inspired hexapod robot control, in: Proceedings of the Fifth International Symposium on Robotics and Manufacturing: Research, Education and Applications, Maui, Hawaii, August 1994. [12] K.S. Espenschied, R.D. Beer, R.D. Quinn, H.J. Chiel, Biologically based distributed control and local reflexes improve rough terrain locomotion in a hexapod robot, Robotics and Autonomous Systems 18 (1996) 59--64. [13] S. Grillner, The role of muscle stiffness in meeting the changing postural and locomotor requirements for force development by the ankle extensors, Acta Physiologica Scandinavica 86 (1972) 92-108. [14] N.C. Heglund G.A. Cavagna, C.R. Taylor, Mechanical work in terrestrial locomotion: Two basic mechanisms for minimizing energy expenditure, American Journal of Physiology 233 (1977) R243-R261. [15] J.A. Hoffer, S. Andreassen, Factors affecting the gain of the stretch reflex and soleus muscle stiffness in premammillary cats, Society of Neuroscience Abstracts 4 (1978) 937. [16] J.A. Hoffer, S. Andreassen, Regulation of soleus muscle stiffness in premarnmillary cats: Intrinsic and reflex components, Journal of Neurphysiology 45 (1981) 267-285. [ 17] N. Hogan, Control of mechanical impedance of prosthetic joints, Joint Automatic Control Conference, 1980. [18] N. Hogan, Impedance control of a robotic manipulator, Winter Annual Meeting of the ASME, 1981. [ 19] N. Hogan, Control strategies for computer movements derived from physical systems theory, InternationalSymposium on Synergetics, 1985. [20] N. Hogan, Impedance control: An approach to manipulation, Journal of Dynamic Systems, Measurements, and Control 107 (1985). [21] N. Hogan, Mechanical impedance of single- and multi-articular systems, in: J.M. Winters, S.L.-Y. Woo, (Eds.), Multiple Muscle Systems: Biomechan!ics and Movement Organization, Springer, New York, 1990, Chapter 9, pp. 149-164. [22] J.C. Houk, Regulatiola of stiffness by skeletomotor reflexes, Annual Review of Physiology 41 (1979) 99-114. [23] D.R. Humphrey, D.J. Reed, Separate cortical systems for the control of joint movement and joint stiffness: Reciprocal activation and coactivation of antagonist muscles, Advances in Neurology 39 (1983) 347-372. [24] R.M. Kolacinski, Dynamics and control of an antagonistic biomimetic actuator system, Ph.D. Thesis, Case Western Reserve University, Department of Mechanical and Aerospace Engineering, Cleveland, OH, May 1997. [25] R. Kolacinski, W. Lin, Stabilizability of an antagonistic biomimetic actuator system, Proceedings of the American Control Conference, June 1998. [26] R. Kolacinski, R. Quinn, Design and mechanics of an antagonistic biomimetic actuator system, Proceedings of the International Conference on Robotics and Automation, May 1998. [27] T. McGeer, Passive dynamic walking, The International Journal of Robotics Research 9 (2) (1990) 62-82. [28] S. Mochon, T.A. Mclviahon, Ballistic walking, Journal of Biomechanics 13 (1980) 49-57. [29] S. Mochon, T.A. Mcldahon, Ballistic walking: An improved model, Mathematical Bin-sciences 52 (1981) 241-260. [30] J.B. Morrell, Parallel coupled micro-macro actuators, AI Technical Report 1563, M1T AI Laboratory, Cambridge, MA, March 1996. [31] G.A. Pratt et al., Stiffness isn't everything, in: Proceedings of Fourth International Symposium on Experimental Robotics, Stanford, CA, 30 June-2 July 1995. [32] G.A. Pratt, M.M. Williamson, Series elastic actuators, in: Proceedings of IROS, Pittsburgh, PA, 1994. [33] M.H. Raibert, Trotting, pacing, and bounding by a quardruped robot, Journal of Biomechanics 23 (suppl. 1) (1990) 79-98. [34] S. Tsuto, S. Sugano, I. Kato, Force control of the robot finger joint equipped with mechanical compliance adjuster, in: Proceedings of 1992 International Conference on Intellegent Robots and Systems, IEEE/RSJ, 1992, pp. 2005-2013.
Richard Kolacinski is an NIH Post-Doctoral Fellow in Case Western Reserve University (CWRU) School of Medicine and a lecturer in the Department of Mechanical and Aerospace Engineering Department of CWRU. He received a Ph.D. degree in Mechanical Engineering from Case Western Reserve University, Cleveland, OH in 1997. He is performing research in biologically inspired robotics and the use of nonlinear control techniques to deduce muscle coordination and joint forces in human walking.
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R.M. Kolacinski, R.D. Quinn /Robotics and Autonomous Systems 25 (1998) 1-18
RogerD. Quinn is a professor on the faculty of the Mechanical and Aerospace Engineering Department at Case Western Reserve University, having joined the department in 1986 as the General Motors Assistant Professor. He received a Ph.D. degree in Engineering Science and Mechanics from Virginia Polytechnic Institute and State University, Blacksburg, in 1985. He has directed the Biorobotics Laboratory at CWRU since its inception in 1991: three insect-inspired robots and one worm inspired robot have been developed. He is also team leader of the Agile Manufacturing Project in the Center for Automation and Intelligent Systems Research at CWRU. His research is devoted to robotics for manufacturing, biologically inspired robotics, and multibody dynamics.
