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Adaptive Tuning of Stiffness and Motion for Multi-Joint Robot*
Adaptive Tuning of Stiffness and Motion for Multi-Joint Robot ⋆ Mitsunori Uemura Sadao Kawamura
Department of Robotics, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, Japan {uemura-m, kawamura}@se.ritsumei.ac.jp
Abstract: This paper proposes a new controller that adaptively tunes joint stiffness and motions of multi-joint robots to reduce actuator torque while generating periodic motions. This controller is composed of delayed feedback control, a stiffness adaptation technique, and positive linear state feedback of angular velocity. Advantages of the proposed controller are to work without exact parameter values of the robots nor huge numerical calculations. We analyzed stability of the closed loop systems. Simulation results show that the proposed controller generate energy saving periodic motions. Additionally, we discuss energy saving effect of linear state feedback of the proposed controller. Keywords: Resonance, Adaptive Control, Optimal Control, Impedance Control, Energy Control. 1. INTRODUCTION
to work well without using exact parameter values of the controlled systems, and low computational cost. We 1.1 Energy Saving Control Method for Multi-Joint Structure mathematically proved stability of some of the controlled systems. In order to reduce energy consumption while generat- In some of these control methods (3; 5), desired motions ing motions, optimal control theory is used traditionally. are specified and fixed. Then, the stiffness adjustment However, for nonlinear optimal control problems, we need reduces actuator torque as much as possible while realizing precise models and use exact parameter values of the trajectory tracking control. However, in these control controlled systems. After the modeling, huge numerical methods, motion optimization is not considered. calculation is required to obtain optimal actuator torque. Then, in many cases, the controlled systems may not be In another our controller (4), not only stiffness but also able to adapt to changes of environments. Moreover, in motions are adjusted to reduce actuator torque by using the conventional optimal control theory, tuning of param- delayed feedback control (DFC) (6). Then, simulation reeters of passive elements is not considered explicitly, even sults showed that the actuator torque almost converged to though some living things seem to tune stiffness of elastic 0 while generating periodic motions of multi-joint robots. tendons or muscles to generate energy saving motions, such In this case, we ignore friction. Therefore, optimal motions can be easily determined as motions, which require no acas hopping motions of kangaroos. tuator torque, like the cases of the passive walking robots. Some researchers tried to generate energy saving motions However, in the case that friction or impact phenomena by using characteristics of Hamiltonian dynamics (1; 2). In consume energy, we have to consider how to generate this case, controllers do not require exact parameter values energy saving motions with using actuator torque. of the controlled systems. However, these control methods can generate only time symmetric motions, and can not 1.3 This Paper generate non symmetric motions. 1.2 Control Method Utilizing Stiffness Optimization To utilize mechanical elastic elements is also effective for robotic systems to generate motions while reducing actuator torque. From this point of view, we have proposed some control methods to generate periodic motions while optimizing stiffness of elastic elements installed in robotic systems (3; 4; 5). In other words, the proposed control method simultaneously utilizes passive elements and actuators. Advantages of the proposed control methods are ⋆ Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.
This paper tries to generate energy saving periodic motions of multi-joint robots with using actuator torque. For this purpose, we propose a new controller that is composed of delayed feedback control, stiffness adaptation and posMulti-Joint Robot
Adjustable Elastic Elements Actuators
Fig. 1. Multi-Joint Robot with Elastic Element
itive linear state feedback of angular velocity. This controller also works without using exact parameter values of the controlled systems nor huge numerical calculations. We analyze stability of the closed loop systems. We conduct some simulations to demonstrate energy saving effect of the proposed controller. Additionally, we mathematically discuss energy saving effect of linear state feedback of angular velocity, which is newly introduced to our controller.
where Kv = diag(kv1 · · · kvn ) is a matrix of a velocity error feedback gains, ∆q˙ = q˙ − q˙ d , q d ∈ ℜn is a desired motion, ˆ = diag(ˆ A a1 , a ˆ2 , · · · a ˆn ), k = (k1 , k2 · · · kn )T , k ∈ ℜn×n is a positive definite gain matrix of stiffness adaptation, ¯ = diag(q1 − qe1 · · · qn − qen ), a ˆ = (ˆ Q a1 , a ˆ2 , · · · a ˆn )T , n×n is a positive definite gain matrix of adaptive a ∈ ℜ ˙ = diag(q˙1 · · · q˙n ), α ∈ ℜ is a scalar parameter tuning, Q constant, which is set from 0 to 1, and T is a cycle time.
2. PROBLEM FORMULATION
We should specify the desired motion in the first cycle q d (t) (0 ≤ t < T ). After the first cycle T ≤ t, the desired motion is adjusted by the equation (5).
This section describes a problem of this study. The controlled systems in this study are multi-joint robots with mechanically adjustable elastic elements as shown in Fig.1. 2.1 Dynamics Dynamics of the multi-joint robots is given by { } 1 ˙ ˙ + D q˙ + g(q) R(q)¨ q+ R(q) + S(q, q) 2 = −K(q − q e ) + τ ,
(1)
where R(q) ∈ ℜn×n is a positive definite inertia matrix, ˙ ∈ ℜn×n n is the number of the robot joints, S(q, q) is a skew symmetric matrix, g(q) ∈ ℜn is a vector of gravitational torque, q = (q1 · · · qn )T is a vector of joint angles, q e = (qe1 · · · qen )T is a vector of equilibrium angles of the elastic elements, τ = (τ1 · · · τn )T is a vector of actuator torque, K = diag(k1 · · · kn ) is a matrix of joint stiffness of the elastic elements, ki > 0 (i = 1 · · · n) are the stiffness. 2.2 Control Objective The control objective in this study is to generate periodic motions while reducing actuator torque as much as possible. 3. CONTROL METHOD In this section, we design a new controller to realize the control objective. In some recent studies, it has been addressed that not to exert braking torque from actuators, which dissipates energy of the multi-joint robots, is effective to reduce actuator torque while generating motions (7; 8). Therefore, we newly introduce positive linear state feedback of the angular velocity Aq˙ to our controller. because work of Aq˙ becomes always positive. Therefore, we combine Aq˙ with our stiffness adaptation controller with delayed feedback control (4). The controller in this section works without using exact parameter values of the controlled systems. 3.1 Controller We propose the following controller.
3.2 Physical Meaning of Controller The actuator torque of the equation (2) is designed using velocity error feedback −Kv ∆q˙ and state feedback of ˆ q. ˙ If the motion q˙ converge to the angular velocity A ˆ will converge to a the desired one q˙ d , the matrix A constant one. Then, the actuator torque will be linear ˆ q. ˙ The stiffness state feedback of the angular velocity A K is adjusted by using the parameter tuning structure of adaptive control as show in the equation (3). The structure of the equation (5) and (3) is almost the same as our previous controller (4), and we have showed that this controller design is effective to generate periodic motions while reducing actuator torque. An equilibrium trajectory of the proposed controller is q˙ = q˙ d . We discuss stability around the equilibrium trajectory in the next section. Then, we can expect the convergence of q˙ to q˙ d , and the actuator torque converges ˙ This means the actuator torque will do only to τ = Aq. positive work. 3.3 Application to Walking Motion Dynamics of walking robots are very complicated, because there are impact and falling phenomena. These phenomena make controller design very difficult. However, when we focus on the period except the moment of the impact phenomena and ignore the falling phenomena, the dynamics becomes the same as multijoint robots (the equation (1)). In this case, to generate motions during the period from just after toe off (initial conditions) to just before heel strike (terminal condition) by using less actuator torque is also the same as the control problem in this study. Therefore, at least for the walking motions except the falling and the impact phenomena, our proposed controller may play a important role. 4. STABILITY ANALYSIS This section discusses stability of the controlled systems. We have not succeeded in proving global stability, and we separately prove local stability of delayed feedback control and stiffness adaptation as an initial step of the stability analysis.
ˆ q˙ τ = −Kv ∆q˙ + A ¯ q˙ k˙ = k Q∆
(2) (3)
4.1 Existence of Energy Saving Motion
a ˆ˙ = −
(4)
Before the stability analysis, we need to assume exsistence of energy saving periodic motions q (t) = q (t − T ), that
˙ ˙ a Q∆q
˙ − T ), q˙ d (t) = (1 − α)q˙ d (t − T ) − αq(t
(5)
are generated by only positive linear state feedback of angular velocity. Then, q satisfies the following equation. { } 1 ˙ R(q )¨ q + R(q ) + S(q , q˙ ) + D q˙ + K q 2 = Aq˙ , (6)
+T Ni = ||∆q˙ d (t)||iT,iT + V (iT + T ). Kv
(14)
Then, the equation (13) can be rewritten as Ni = Ni
1
˙ iT − (2α − α2 )||∆q|| Kv
T,iT
(15)
.
where K = diag(kopt1 , kopt2 , · · · koptn ) ∈ ℜn×n , A = diag(a1 , a2 , · · · an ) ∈ ℜn×n are constant stiffness matrixes.
Therefore, when we set α from 0 to 2, the series Ni decreases every cycle. This means q˙ → q˙ d , and brings about q˙ d → q˙ in the sense of L2 norm.
4.2 Stability of Delayed Feedback Control
Therefore, we could prove local stability of the controlled systems with delayed feedback control.
In this part, we show local stability of delayed feedback control. We can obtain an error equation between the desired motion q˙ d and the energy saving motion q˙ by subtracting q˙ from both side of the the equation (5), ˙ − T) ∆q˙ d (t) = ∆q˙ d (t − T ) + α∆q(t
(7)
where ∆q˙ d = q˙ d − q˙ . Then, we can obtain the following norm by taking inner product of both side of the equation (7) with multiplying Kv . ˙ − T )||Kv ||∆q˙ d (t)||Kv = ||∆q˙ d (t − T )||Kv + α2 ||∆q(t ˙ − T ), +2α∆q˙ d (t − T )T Kv ∆q(t
(8)
where we used the form of the norm ||∆q˙ d (t)||Kv = ∆q˙ d (t)T Kv ∆q˙ d (t). Next, we define the following scalar function V (t). V (t) =
1 1 ∆q˙ T R(q )∆q˙ + ∆q T K ∆q ≥ 0 2 2
(9)
In this paper, we analyze stability around the optimal motion q(t) → q (t) the optimal constant stiffness K → ˆ → A. Therefore, we consider the following K , and A linearlized dynamics. { } 1 ˙ R(q )¨ q+ R(q ) + S(q , q˙ ) q˙ + K q = 2 −Kv ∆q˙ − Kp ∆q + Aq˙ (10) Then, we can obtain time derivative of the equation (9) as ˙ Kv − ∆q˙ T Kv ∆q˙ d . (11) V˙ (t) = −∆q˙ T Kv ∆q˙ = −||∆q|| From the equation (8) and the equation (11), we obtain ||∆q˙ d (t)||Kv = ||∆q˙ d (t − T )||Kv ˙ − T )||Kv − V˙ (t). (12) −(2α − α2 )||∆q(t By integrating the equation (12) with respect to time t from t to t + T , we obtain ||∆q˙ d (t)||t,t+T + V (t + T ) Kv ˙ − T )||t,t+T = ||∆q˙ d (t − T )||t,t+T − (2α − α2 )||∆q(t Kv Kv +V (t)
(13)
where we used the form of the norm ||∆q˙ d (t)||t,t+T = Kv ∫ t+T T ∆q˙ d (t) Kv ∆q˙ d (t)dt. Next, we define the following t scalar series Ni (i = 1, 2, 3, · · · )
4.3 Stability of Stiffness Adaptation This section proves local stability of the controlled systems with the adaptive parameter tuning in the proposed controller. In this analysis, we assume that the desired motion already converges to the energy saving motion q˙ d → q˙ . Then, the actuator torque τ will be τ = −Kv ∆q˙ − ˆ q. ˙ Under the assumption, we define the folKp ∆q + A lowing candidate of Lyapnov function V . 1 1 V2 (t) = ∆q˙ T R(q)∆q˙ + ∆q T (K + Kp − βKv )∆q 2 2 1 T +β∆q˙ R(q)∆q + ∆kT k 1 ∆k 2 1 1 T + ∆a a ∆a ≥ 0 (16) 2 By using the equation (6) and discussion of the book (9), we obtain ˙ Kv V˙ 2 (t) < −||∆q||
ˆ c1 I A
− ||∆q ||βKp
c2 I ,
(17)
where c1 and c2 are positive constants. These constants c1 , c2 are independent of the feedback gains Kv , Kp . Therefore, if we set enough large gains Kv , Kp , the V2 (t) decreases monotonically. Then, we can prove convergences ˆ → A. of the state variables q → q , K → K , A 4.4 Summary The above analyses imply the following two ability of the proposed controller. Delayed feedback control of the equation (5) have the ability of convergence of the desired motion q˙ d to the energy saving motions q˙ . The parameter adaptation laws of the equation (3) and (4) and error feedback −Kv ∆q˙ − Kp ∆q of the equation (2) have the ability of convergence of the motion q to the desired motion q d . Therefore, the combination of these schemes seems to enable the convergence of the actuator torque τ to positive ˆ q. ˙ However, in linear state feedback of angular velocity A this analysis, some problems still remain, because we used some assumptions. To prove global stability without using these assumptions is our important future work. 5. SIMULATION This section confirms the effectiveness of the proposed controller through some simulation results.
5.1 Model A planar 2 link walking robot as shown in Fig.2 is adopted as the simulation model. In the case of actual walking robots, falling phenomena will occur if heels or toes of stance legs will be away from the ground. However, in this simulation, we ignore the falling phenomena to focus on generating energy saving motions. The foot of the swinging leg is assumed to be small enough to ignore its dynamics. Dynamics of the simulation model is assumed to be the equation (1) and physical parameters are set to m1 = 3.0[kg], m2 = 3.0[kg], l1 = 0.25[m], l2 = 0.25[m], lg1 = 0.15[m], lg2 = 0.1[m], I1 = 0.01[Nms2 /rad], I2 = 0.01[Nms2 /rad]. The m is weight of the robot link, l is length of the link, lg is length from the joint to the mass center of the link, I is inertia moment around the mass center of the link, The suffixes 1 ,2 represent the parameters of 1, 2th link. The impact phenomena are assumed to be perfectly inelastic collisions. 5.2 Condition We conduct two cases of simulations, and we call them ”case 1” and ”case 2”. In the case 1, we set a sinusoidal desired motion qd1 = −0.5 cos(1.5πt) + qe1 [rad], qd2 = 1.1 cos(1.8πt + 0.04π) + qe2 [rad] to each joint, and actuator torque is designed by only velocity feedback control τ = Kv (q˙ − q˙ d ). The gains are set to Kv = diag(3.4, 0.5)[Nms/rad]. In this case, we do not use the elastic elements, and k1 = k2 = 0[Nm/rad]. In the case 2, we adopt the proposed controller of the equations from (2) to (5). The gains are set to Kv = diag(3.4, 0.5)[Nms/rad], k = diag(10, 0.4), a = diag(0.02, 0.005), α = 0.5[-]. Initial desired motions q d (t) (0 ≤ t < T ) are set to qd1 = −0.5 cos(1.5πt) + qe1 [rad], qd2 = 1.1 cos(1.8πt + 0.04π)+qe2 [rad]. Initial stiffness are set to k1 = 0[Nm/rad], k2 = 0[Nm/rad], and their equilibrium angles are set to qe1 = 21 π[rad], and qe2 = −π[rad]. 5.3 Result
actuator torque of the proposed controller became almost proportional to the angular velocity τ1 = a ˆ1 q˙1 , τ2 = a ˆ2 q˙2 as shown in Fig.5(a), (b). We evaluated the saving effect by using the cost ∫ tenergy end function Jsim = tstart τ T τ dt, where the tstart , tend is the time. When we set the time period as tstart = 10 and tend = 15, the cost function became 20.4[N2 m2 s] in the case 1. In the case 2, we set the time period as tstart = 295 and tend = 300, and we obtain the cost function as 1.07[N2 m2 s]. Therefore, we could reduce actuator torque more than 90 [%] by the proposed controller. The above results demonstrated that the proposed controller generated the energy saving motion. When we changed the gain values, we could obtain similar results, even speed of convergence is different. To increase the velocity feedback gains Kv enhances the convergence speed of the motion q˙ to the desired one q˙ d . However, in such cases, the motion modification by the equation (5) became slower because the difference between the two ˙ q˙ d was reduced. To increase the value α of motions q, the equation (5) enhances the modification speed of the desired motion q˙ d . However, to increase the α too much brings about large differences between the initial desired motions and the converged desired motions. Therefore, the values of α and Kv should not be too small nor too large. The balance between the feedback gains Kv and the adaptive gains k , a for the convergence speed was almost the same as the balance between those of usual adaptive controllers. To investigate the good balances of these gains is our future work. 6. DISCUSSION ON POSITIVE LINEAR STATE FEEDBACK OF ANGULAR VELOCITY In our proposed controller, we used positive linear state ˙ In this section, we discuss feedback of angular velocity Aq. an energy saving effect of linear state feedback in detail. 6.1 Torque Minimization Problem
where A = A
q1
Fig. 2. Simulation model
(A − D)A
1
.
(a)
q1 2.0
(b)
1.5 1.0
τ1
1
-1
0
10
Time[s]
(d)
Angle [rad]
In this analysis, we consider the cases that a1 , a2 , · · · an are larger than d1 , d2 , · · · dn respectively. We consider fixed
(c)
τ1
T
Torque [Nm]
q2
0
′
Angle [rad]
τ2
Let us consider torque minimization problem of the multijoint robots, and introduce the following cost function ∫ T ′ J= τ A τ dt, (18)
Torque [Nm]
Simulation results are shown in Fig.3, Fig.4 and Fig.5. As shown in Fig.3(a), (b) and Fig.4(a), (b), motions are converged to periodic patterns after some cycles. These converged motions are not so different from each other cases. The parameters k1 , k2 , a ˆ1 and a ˆ2 converged to some constants as shown in Fig.4(c) and Fig.4(d). As the result, the actuator torque of the proposed controller became much smaller than the velocity feedback controller as shown in Fig.3(c), (d), and Fig.4(e), (f ). After about 300 seconds of the simulation of the case 2, the
q2
-2 -3 -4
τ2
1
-1
0
10
Time[s]
Fig. 3. Simulation Results of Case 1 (Velocity Feedback Controller with Sinusoidal Desired Motion)
(c)
Angle [rad]
1.5 1.0
Angle [rad]
(b)
2.0
-2
Stiffness [Nm/rad]
(a)
3
-3 -4
k1 k2
[Nms/rad]
0
(f)
0.02
a1 a2
0.00 5
τ1
0
-5
Torque [Nm]
(e)
Torque [Nm]
(d)
τ2
1 0 -1
0
10
20
30
40
50
60
70
80
90
100
Time[s]
Fig. 4. Simulation Results of Case 2 (Proposed Controller) ˙ initial conditions q(0) = qstart , q(0) = vstart when t = 0, ˙ ) = vend when and terminal conditions q(T ) = qend , q(T t = T . If the motions are periodic q(t) = q(t + T ) and there is no jump of state variables, initial and terminal conditions will be the same qstart = qend , vstart = vend . However, in this analysis, we don’t restrict the initial and the terminal conditions to be the same.
It is well known that robot dynamics satisfies the energy ∫T ∫T ˙ relationship 0 q˙ T τ dt = E(T )−E(0)+ 0 q˙ T D qdt, where E(t) ∈ ℜ is total energy of the multi-joint robots. Since the initial (t = 0) and the terminal (t = T ) conditions are fixed, we can introduce a scalar constant c = E(T ) − E(0). Then, the energy relationship can be rewritten due to the ˙ orthogonality of τc to q.
6.2 Analysis on torque minimization problem
∫
Here, we analytically derive that the optimal actuator torque τ opt , which minimizes the cost function J, can be described as linear state feedback of the angular velocity ˙ q. At first, we decompose the actuator torque τ to a component proportional to the angular velocity q˙ and a remain˙ Then, ing component τc = (τc1 , τc2 , · · · τcn )T = τ − Aq. the actuator torque can be described by τ = Aq˙ + τc . The coefficients ai (i = 1, 2, · · · n) of the component of the ∫ T
τi q˙i dt
angular velocity are calculated by ai = ∫0 T 0
q˙i2 dt
. There-
0.001
τ1 (b) a1
0.000
295
300
Time[s]
Torque [Nm]
(a)
Torque [Nm]
fore, the remaining component τci does not include the component of the angular velocity q˙i , and inner product ∫T between the two components 0 q˙i τci dt will be 0, because the two components have a orthogonal relationship. 0.04
τ2 a2
0.00 -0.04
T
q˙ τ dt = T
0
∫
T 0
˙ 0,T ˙ 0,T q˙ T (Aq˙ + τc )dt = ||q|| A = c + ||q|| D (19)
Based on the above energy analysis, we derive the optimal actuator torque τ opt as follows. First, we substitute the decomposed components of the actuator torque τ = Aq˙ + τc into the cost function J. J=
∫
T 0
{
} ′ ′ q˙ T (A − D)q˙ + 2q˙ T AT A τc + τc T A τc dt (20)
By using the orthogonality of τci to q˙i and the energy relationship of the equation (19), we obtain J =c+
∫
T
′
τc T A τc dt.
(21)
0
Obviously, the cost function J will be the smallest when τc = 0. Therefore, the optimal actuator torque τ opt can ˙ be described by the linear state feedback form τ opt = Aq. 6.3 Discussion
295
300
Time[s]
Fig. 5. Converged Actuator Torque of Case 2
Physical meaning of the above analysis is that minimum actuator torque is required to generate motions without
exerting torque, which does not contribute the energy supply. 6.4 Analysis Based on Hamilton-Jacobi-Bellman Equation We can lead to similar conclusion τ opt = Aq˙ by using the Hamilton-Jacobi-Bellman equation. This kind of analysis by using Hamilton-Jacobi-Bellman equation is proposed by Arimoto et al.(10). However, Arimoto et al.considered only a damping shaping problem of position control, and they did not consider problems of motion generation. In this analysis, we ignore the viscosity D = 0, and we ∫T set the cost function as J = 0 τ A 1 τ dt instead of the equation (18). Then, the Hamiltonian [ { is given by H} = ˙ ˙ q˙ + S(q, q) τ T A 1 τ + λ1 T p + λ2 T R(q) 1 − 12 R(q) ˙ −g(q) − K(q − q e ) + τ ], where p = q. We can obtain the conclusion τ opt = Aq˙ by using the analysis of the paper (10). A difference from the paper (10) is to set the value function v to −E(t). Therefore, it is proved by using the Hamilton-Jacobi-Bellman equation that the optimal control can be described by linear state feedback. 6.5 Comparison of Proposed Formulation with Conventional Resonance Next, we discuss analogies and differences between conventional resonance and the discussion in this paper. Conventional resonance is formulated for 1-DOF linear dynamics m¨ q + dq˙ + kq = τ where m ∈ ℜ is an inertia coefficient, d ∈ ℜ is a viscosity, k ∈ ℜ is stiffness, q ∈ ℜ is displacement, and τ ∈ ℜ is actuator torque. In the resonant condition, motions will be sinusoidal q˙ = ad sin(ωt + ϕ), and stiffness k satisfies the condition k = mω 2 . Then, the dynamics will be equivalent to a pure viscous system dq˙ = τ , and the actuator torque will be linear state feedback of the velocity q. ˙ In the resonant condition, the periodic motions q(t) = q(t + T ) ∫T are generated by minimum actuator torque 0 τ 2 dt. In our discussion, the actuator torque is linear state ˙ Therefore, the feedback of the angular velocity τ = Aq. property of linear state feedback is the same between the proposed formulation and conventional resonance. Generating motions by minimum actuator torque is also the same. Therefore, our proposed formulation can be regarded as a kind of extension of resonance to the periodic motions of the multi-joint robots. 7. CONCLUSION This paper proposes a new controller to realize an effective simultaneous use of passive elements and actuators . This proposed controller is composed of adaptive parameter tunings, delayed feedback control, and linear state feedback of angular velocity. Advantages of our proposed controller are to work without using exact parameter values of the controlled system nor huge numerical calculation. We analyzed that the closed loop systems are locally stable.
Simulation results demonstrated the effectiveness of the proposed controller and proposed resonance. Additionally, we discuss an energy saving effect of linear state feedback of angular velocity. Then, a minimum torque controller can be represented as linear state feed˙ Physical meaning of this property is that back τ = Aq. minimum torque is required to generate periodic motions without exerting torque, which does not contribute energy supply. This property is the same as conventional resonance. To prove global stability of the controlled systems with the proposed controller is our important future work. 8. ACKNOWLEDGMENT This work has been partially supported by a Grantin-Aid for Young Scientists (Start-up) ”An Extension of Resonance to Multi-Joint Structures and Adaptive Control Methods Based on Extended Resonance” from the Japanese Ministry of Education, Culture, Sports, Science and Technology. REFERENCES [1] S. Satoh, K. Fujimoto, S. Hyon: “Gait generation for passive running via iterative learning control”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2006, pp. 5907-5912, 2008. [2] S. Hyon, T. Emura: “Symmetric walking control: Invariance and global stability”, IEEE International Conference on Robotics and Automation 2005, pp. 1455-1462, 2005. [3] M. Uemura, S. Kawamura: “An Energy Saving Control Method of Robot Motions based on Adaptive Stiffness Optimization - Cases of Multi-Frequency Components -”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2008, pp. 551-557, 2008. [4] M. Uemura, G. Lu, S. Kawamura, S. Ma: “Passive Periodic Motions of Multi-Joint Robots by Stiffness Adaptation and DFC for Energy Saving”, SICE Annual Conference 2008, pp. 2853-2858, 2008. [5] M. Uemura, S. Kawamura: “Resonance-based Motion Control Method for Multi-Joint Robot through Combining Stiffness Adaptation and Iterative Learning Control”, IEEE International Conference on Robotics and Automation 2009, pp. 1543-1548, 2009. [6] K. Pyragas: “Continuous Control of Chaos by SelfControlling Feedback”, Physics Letters A, Vol. 170, No. 6, pp. 412-428, 1992. [7] S. Collins, A. Ruina, R. Tedrake, M. Wisse: “Efficient bipedal robots based on passive-dynamic walkers,” Science, vol. 307, pp. 1082-1085, 2005. [8] F. Asano, Z.W. Luo, M. Yamakita: “Unification of Dynamic Gait Generation Methods via Variable Virtual Gravity and Its Control Performance Analysis”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2004, pp. 3865-3870, 2004. [9] S. Arimoto: Control Theory of Non-Linear Mechanical Systems: A Passivity-Based and Circuit-Theoretic Approach. Oxford Engineering Science Series, 1996. [10] S. Arimoto: “Extension of Impedance Matching to Nonlinear Dynamics of Robotic Tasks”, System and Control Letters, Vol. 36, No. 2, pp. 109-119, 1998.
Department of Robotics, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, Japan {uemura-m, kawamura}@se.ritsumei.ac.jp
Abstract: This paper proposes a new controller that adaptively tunes joint stiffness and motions of multi-joint robots to reduce actuator torque while generating periodic motions. This controller is composed of delayed feedback control, a stiffness adaptation technique, and positive linear state feedback of angular velocity. Advantages of the proposed controller are to work without exact parameter values of the robots nor huge numerical calculations. We analyzed stability of the closed loop systems. Simulation results show that the proposed controller generate energy saving periodic motions. Additionally, we discuss energy saving effect of linear state feedback of the proposed controller. Keywords: Resonance, Adaptive Control, Optimal Control, Impedance Control, Energy Control. 1. INTRODUCTION
to work well without using exact parameter values of the controlled systems, and low computational cost. We 1.1 Energy Saving Control Method for Multi-Joint Structure mathematically proved stability of some of the controlled systems. In order to reduce energy consumption while generat- In some of these control methods (3; 5), desired motions ing motions, optimal control theory is used traditionally. are specified and fixed. Then, the stiffness adjustment However, for nonlinear optimal control problems, we need reduces actuator torque as much as possible while realizing precise models and use exact parameter values of the trajectory tracking control. However, in these control controlled systems. After the modeling, huge numerical methods, motion optimization is not considered. calculation is required to obtain optimal actuator torque. Then, in many cases, the controlled systems may not be In another our controller (4), not only stiffness but also able to adapt to changes of environments. Moreover, in motions are adjusted to reduce actuator torque by using the conventional optimal control theory, tuning of param- delayed feedback control (DFC) (6). Then, simulation reeters of passive elements is not considered explicitly, even sults showed that the actuator torque almost converged to though some living things seem to tune stiffness of elastic 0 while generating periodic motions of multi-joint robots. tendons or muscles to generate energy saving motions, such In this case, we ignore friction. Therefore, optimal motions can be easily determined as motions, which require no acas hopping motions of kangaroos. tuator torque, like the cases of the passive walking robots. Some researchers tried to generate energy saving motions However, in the case that friction or impact phenomena by using characteristics of Hamiltonian dynamics (1; 2). In consume energy, we have to consider how to generate this case, controllers do not require exact parameter values energy saving motions with using actuator torque. of the controlled systems. However, these control methods can generate only time symmetric motions, and can not 1.3 This Paper generate non symmetric motions. 1.2 Control Method Utilizing Stiffness Optimization To utilize mechanical elastic elements is also effective for robotic systems to generate motions while reducing actuator torque. From this point of view, we have proposed some control methods to generate periodic motions while optimizing stiffness of elastic elements installed in robotic systems (3; 4; 5). In other words, the proposed control method simultaneously utilizes passive elements and actuators. Advantages of the proposed control methods are ⋆ Sponsor and financial support acknowledgment goes here. Paper titles should be written in uppercase and lowercase letters, not all uppercase.
This paper tries to generate energy saving periodic motions of multi-joint robots with using actuator torque. For this purpose, we propose a new controller that is composed of delayed feedback control, stiffness adaptation and posMulti-Joint Robot
Adjustable Elastic Elements Actuators
Fig. 1. Multi-Joint Robot with Elastic Element
itive linear state feedback of angular velocity. This controller also works without using exact parameter values of the controlled systems nor huge numerical calculations. We analyze stability of the closed loop systems. We conduct some simulations to demonstrate energy saving effect of the proposed controller. Additionally, we mathematically discuss energy saving effect of linear state feedback of angular velocity, which is newly introduced to our controller.
where Kv = diag(kv1 · · · kvn ) is a matrix of a velocity error feedback gains, ∆q˙ = q˙ − q˙ d , q d ∈ ℜn is a desired motion, ˆ = diag(ˆ A a1 , a ˆ2 , · · · a ˆn ), k = (k1 , k2 · · · kn )T , k ∈ ℜn×n is a positive definite gain matrix of stiffness adaptation, ¯ = diag(q1 − qe1 · · · qn − qen ), a ˆ = (ˆ Q a1 , a ˆ2 , · · · a ˆn )T , n×n is a positive definite gain matrix of adaptive a ∈ ℜ ˙ = diag(q˙1 · · · q˙n ), α ∈ ℜ is a scalar parameter tuning, Q constant, which is set from 0 to 1, and T is a cycle time.
2. PROBLEM FORMULATION
We should specify the desired motion in the first cycle q d (t) (0 ≤ t < T ). After the first cycle T ≤ t, the desired motion is adjusted by the equation (5).
This section describes a problem of this study. The controlled systems in this study are multi-joint robots with mechanically adjustable elastic elements as shown in Fig.1. 2.1 Dynamics Dynamics of the multi-joint robots is given by { } 1 ˙ ˙ + D q˙ + g(q) R(q)¨ q+ R(q) + S(q, q) 2 = −K(q − q e ) + τ ,
(1)
where R(q) ∈ ℜn×n is a positive definite inertia matrix, ˙ ∈ ℜn×n n is the number of the robot joints, S(q, q) is a skew symmetric matrix, g(q) ∈ ℜn is a vector of gravitational torque, q = (q1 · · · qn )T is a vector of joint angles, q e = (qe1 · · · qen )T is a vector of equilibrium angles of the elastic elements, τ = (τ1 · · · τn )T is a vector of actuator torque, K = diag(k1 · · · kn ) is a matrix of joint stiffness of the elastic elements, ki > 0 (i = 1 · · · n) are the stiffness. 2.2 Control Objective The control objective in this study is to generate periodic motions while reducing actuator torque as much as possible. 3. CONTROL METHOD In this section, we design a new controller to realize the control objective. In some recent studies, it has been addressed that not to exert braking torque from actuators, which dissipates energy of the multi-joint robots, is effective to reduce actuator torque while generating motions (7; 8). Therefore, we newly introduce positive linear state feedback of the angular velocity Aq˙ to our controller. because work of Aq˙ becomes always positive. Therefore, we combine Aq˙ with our stiffness adaptation controller with delayed feedback control (4). The controller in this section works without using exact parameter values of the controlled systems. 3.1 Controller We propose the following controller.
3.2 Physical Meaning of Controller The actuator torque of the equation (2) is designed using velocity error feedback −Kv ∆q˙ and state feedback of ˆ q. ˙ If the motion q˙ converge to the angular velocity A ˆ will converge to a the desired one q˙ d , the matrix A constant one. Then, the actuator torque will be linear ˆ q. ˙ The stiffness state feedback of the angular velocity A K is adjusted by using the parameter tuning structure of adaptive control as show in the equation (3). The structure of the equation (5) and (3) is almost the same as our previous controller (4), and we have showed that this controller design is effective to generate periodic motions while reducing actuator torque. An equilibrium trajectory of the proposed controller is q˙ = q˙ d . We discuss stability around the equilibrium trajectory in the next section. Then, we can expect the convergence of q˙ to q˙ d , and the actuator torque converges ˙ This means the actuator torque will do only to τ = Aq. positive work. 3.3 Application to Walking Motion Dynamics of walking robots are very complicated, because there are impact and falling phenomena. These phenomena make controller design very difficult. However, when we focus on the period except the moment of the impact phenomena and ignore the falling phenomena, the dynamics becomes the same as multijoint robots (the equation (1)). In this case, to generate motions during the period from just after toe off (initial conditions) to just before heel strike (terminal condition) by using less actuator torque is also the same as the control problem in this study. Therefore, at least for the walking motions except the falling and the impact phenomena, our proposed controller may play a important role. 4. STABILITY ANALYSIS This section discusses stability of the controlled systems. We have not succeeded in proving global stability, and we separately prove local stability of delayed feedback control and stiffness adaptation as an initial step of the stability analysis.
ˆ q˙ τ = −Kv ∆q˙ + A ¯ q˙ k˙ = k Q∆
(2) (3)
4.1 Existence of Energy Saving Motion
a ˆ˙ = −
(4)
Before the stability analysis, we need to assume exsistence of energy saving periodic motions q (t) = q (t − T ), that
˙ ˙ a Q∆q
˙ − T ), q˙ d (t) = (1 − α)q˙ d (t − T ) − αq(t
(5)
are generated by only positive linear state feedback of angular velocity. Then, q satisfies the following equation. { } 1 ˙ R(q )¨ q + R(q ) + S(q , q˙ ) + D q˙ + K q 2 = Aq˙ , (6)
+T Ni = ||∆q˙ d (t)||iT,iT + V (iT + T ). Kv
(14)
Then, the equation (13) can be rewritten as Ni = Ni
1
˙ iT − (2α − α2 )||∆q|| Kv
T,iT
(15)
.
where K = diag(kopt1 , kopt2 , · · · koptn ) ∈ ℜn×n , A = diag(a1 , a2 , · · · an ) ∈ ℜn×n are constant stiffness matrixes.
Therefore, when we set α from 0 to 2, the series Ni decreases every cycle. This means q˙ → q˙ d , and brings about q˙ d → q˙ in the sense of L2 norm.
4.2 Stability of Delayed Feedback Control
Therefore, we could prove local stability of the controlled systems with delayed feedback control.
In this part, we show local stability of delayed feedback control. We can obtain an error equation between the desired motion q˙ d and the energy saving motion q˙ by subtracting q˙ from both side of the the equation (5), ˙ − T) ∆q˙ d (t) = ∆q˙ d (t − T ) + α∆q(t
(7)
where ∆q˙ d = q˙ d − q˙ . Then, we can obtain the following norm by taking inner product of both side of the equation (7) with multiplying Kv . ˙ − T )||Kv ||∆q˙ d (t)||Kv = ||∆q˙ d (t − T )||Kv + α2 ||∆q(t ˙ − T ), +2α∆q˙ d (t − T )T Kv ∆q(t
(8)
where we used the form of the norm ||∆q˙ d (t)||Kv = ∆q˙ d (t)T Kv ∆q˙ d (t). Next, we define the following scalar function V (t). V (t) =
1 1 ∆q˙ T R(q )∆q˙ + ∆q T K ∆q ≥ 0 2 2
(9)
In this paper, we analyze stability around the optimal motion q(t) → q (t) the optimal constant stiffness K → ˆ → A. Therefore, we consider the following K , and A linearlized dynamics. { } 1 ˙ R(q )¨ q+ R(q ) + S(q , q˙ ) q˙ + K q = 2 −Kv ∆q˙ − Kp ∆q + Aq˙ (10) Then, we can obtain time derivative of the equation (9) as ˙ Kv − ∆q˙ T Kv ∆q˙ d . (11) V˙ (t) = −∆q˙ T Kv ∆q˙ = −||∆q|| From the equation (8) and the equation (11), we obtain ||∆q˙ d (t)||Kv = ||∆q˙ d (t − T )||Kv ˙ − T )||Kv − V˙ (t). (12) −(2α − α2 )||∆q(t By integrating the equation (12) with respect to time t from t to t + T , we obtain ||∆q˙ d (t)||t,t+T + V (t + T ) Kv ˙ − T )||t,t+T = ||∆q˙ d (t − T )||t,t+T − (2α − α2 )||∆q(t Kv Kv +V (t)
(13)
where we used the form of the norm ||∆q˙ d (t)||t,t+T = Kv ∫ t+T T ∆q˙ d (t) Kv ∆q˙ d (t)dt. Next, we define the following t scalar series Ni (i = 1, 2, 3, · · · )
4.3 Stability of Stiffness Adaptation This section proves local stability of the controlled systems with the adaptive parameter tuning in the proposed controller. In this analysis, we assume that the desired motion already converges to the energy saving motion q˙ d → q˙ . Then, the actuator torque τ will be τ = −Kv ∆q˙ − ˆ q. ˙ Under the assumption, we define the folKp ∆q + A lowing candidate of Lyapnov function V . 1 1 V2 (t) = ∆q˙ T R(q)∆q˙ + ∆q T (K + Kp − βKv )∆q 2 2 1 T +β∆q˙ R(q)∆q + ∆kT k 1 ∆k 2 1 1 T + ∆a a ∆a ≥ 0 (16) 2 By using the equation (6) and discussion of the book (9), we obtain ˙ Kv V˙ 2 (t) < −||∆q||
ˆ c1 I A
− ||∆q ||βKp
c2 I ,
(17)
where c1 and c2 are positive constants. These constants c1 , c2 are independent of the feedback gains Kv , Kp . Therefore, if we set enough large gains Kv , Kp , the V2 (t) decreases monotonically. Then, we can prove convergences ˆ → A. of the state variables q → q , K → K , A 4.4 Summary The above analyses imply the following two ability of the proposed controller. Delayed feedback control of the equation (5) have the ability of convergence of the desired motion q˙ d to the energy saving motions q˙ . The parameter adaptation laws of the equation (3) and (4) and error feedback −Kv ∆q˙ − Kp ∆q of the equation (2) have the ability of convergence of the motion q to the desired motion q d . Therefore, the combination of these schemes seems to enable the convergence of the actuator torque τ to positive ˆ q. ˙ However, in linear state feedback of angular velocity A this analysis, some problems still remain, because we used some assumptions. To prove global stability without using these assumptions is our important future work. 5. SIMULATION This section confirms the effectiveness of the proposed controller through some simulation results.
5.1 Model A planar 2 link walking robot as shown in Fig.2 is adopted as the simulation model. In the case of actual walking robots, falling phenomena will occur if heels or toes of stance legs will be away from the ground. However, in this simulation, we ignore the falling phenomena to focus on generating energy saving motions. The foot of the swinging leg is assumed to be small enough to ignore its dynamics. Dynamics of the simulation model is assumed to be the equation (1) and physical parameters are set to m1 = 3.0[kg], m2 = 3.0[kg], l1 = 0.25[m], l2 = 0.25[m], lg1 = 0.15[m], lg2 = 0.1[m], I1 = 0.01[Nms2 /rad], I2 = 0.01[Nms2 /rad]. The m is weight of the robot link, l is length of the link, lg is length from the joint to the mass center of the link, I is inertia moment around the mass center of the link, The suffixes 1 ,2 represent the parameters of 1, 2th link. The impact phenomena are assumed to be perfectly inelastic collisions. 5.2 Condition We conduct two cases of simulations, and we call them ”case 1” and ”case 2”. In the case 1, we set a sinusoidal desired motion qd1 = −0.5 cos(1.5πt) + qe1 [rad], qd2 = 1.1 cos(1.8πt + 0.04π) + qe2 [rad] to each joint, and actuator torque is designed by only velocity feedback control τ = Kv (q˙ − q˙ d ). The gains are set to Kv = diag(3.4, 0.5)[Nms/rad]. In this case, we do not use the elastic elements, and k1 = k2 = 0[Nm/rad]. In the case 2, we adopt the proposed controller of the equations from (2) to (5). The gains are set to Kv = diag(3.4, 0.5)[Nms/rad], k = diag(10, 0.4), a = diag(0.02, 0.005), α = 0.5[-]. Initial desired motions q d (t) (0 ≤ t < T ) are set to qd1 = −0.5 cos(1.5πt) + qe1 [rad], qd2 = 1.1 cos(1.8πt + 0.04π)+qe2 [rad]. Initial stiffness are set to k1 = 0[Nm/rad], k2 = 0[Nm/rad], and their equilibrium angles are set to qe1 = 21 π[rad], and qe2 = −π[rad]. 5.3 Result
actuator torque of the proposed controller became almost proportional to the angular velocity τ1 = a ˆ1 q˙1 , τ2 = a ˆ2 q˙2 as shown in Fig.5(a), (b). We evaluated the saving effect by using the cost ∫ tenergy end function Jsim = tstart τ T τ dt, where the tstart , tend is the time. When we set the time period as tstart = 10 and tend = 15, the cost function became 20.4[N2 m2 s] in the case 1. In the case 2, we set the time period as tstart = 295 and tend = 300, and we obtain the cost function as 1.07[N2 m2 s]. Therefore, we could reduce actuator torque more than 90 [%] by the proposed controller. The above results demonstrated that the proposed controller generated the energy saving motion. When we changed the gain values, we could obtain similar results, even speed of convergence is different. To increase the velocity feedback gains Kv enhances the convergence speed of the motion q˙ to the desired one q˙ d . However, in such cases, the motion modification by the equation (5) became slower because the difference between the two ˙ q˙ d was reduced. To increase the value α of motions q, the equation (5) enhances the modification speed of the desired motion q˙ d . However, to increase the α too much brings about large differences between the initial desired motions and the converged desired motions. Therefore, the values of α and Kv should not be too small nor too large. The balance between the feedback gains Kv and the adaptive gains k , a for the convergence speed was almost the same as the balance between those of usual adaptive controllers. To investigate the good balances of these gains is our future work. 6. DISCUSSION ON POSITIVE LINEAR STATE FEEDBACK OF ANGULAR VELOCITY In our proposed controller, we used positive linear state ˙ In this section, we discuss feedback of angular velocity Aq. an energy saving effect of linear state feedback in detail. 6.1 Torque Minimization Problem
where A = A
q1
Fig. 2. Simulation model
(A − D)A
1
.
(a)
q1 2.0
(b)
1.5 1.0
τ1
1
-1
0
10
Time[s]
(d)
Angle [rad]
In this analysis, we consider the cases that a1 , a2 , · · · an are larger than d1 , d2 , · · · dn respectively. We consider fixed
(c)
τ1
T
Torque [Nm]
q2
0
′
Angle [rad]
τ2
Let us consider torque minimization problem of the multijoint robots, and introduce the following cost function ∫ T ′ J= τ A τ dt, (18)
Torque [Nm]
Simulation results are shown in Fig.3, Fig.4 and Fig.5. As shown in Fig.3(a), (b) and Fig.4(a), (b), motions are converged to periodic patterns after some cycles. These converged motions are not so different from each other cases. The parameters k1 , k2 , a ˆ1 and a ˆ2 converged to some constants as shown in Fig.4(c) and Fig.4(d). As the result, the actuator torque of the proposed controller became much smaller than the velocity feedback controller as shown in Fig.3(c), (d), and Fig.4(e), (f ). After about 300 seconds of the simulation of the case 2, the
q2
-2 -3 -4
τ2
1
-1
0
10
Time[s]
Fig. 3. Simulation Results of Case 1 (Velocity Feedback Controller with Sinusoidal Desired Motion)
(c)
Angle [rad]
1.5 1.0
Angle [rad]
(b)
2.0
-2
Stiffness [Nm/rad]
(a)
3
-3 -4
k1 k2
[Nms/rad]
0
(f)
0.02
a1 a2
0.00 5
τ1
0
-5
Torque [Nm]
(e)
Torque [Nm]
(d)
τ2
1 0 -1
0
10
20
30
40
50
60
70
80
90
100
Time[s]
Fig. 4. Simulation Results of Case 2 (Proposed Controller) ˙ initial conditions q(0) = qstart , q(0) = vstart when t = 0, ˙ ) = vend when and terminal conditions q(T ) = qend , q(T t = T . If the motions are periodic q(t) = q(t + T ) and there is no jump of state variables, initial and terminal conditions will be the same qstart = qend , vstart = vend . However, in this analysis, we don’t restrict the initial and the terminal conditions to be the same.
It is well known that robot dynamics satisfies the energy ∫T ∫T ˙ relationship 0 q˙ T τ dt = E(T )−E(0)+ 0 q˙ T D qdt, where E(t) ∈ ℜ is total energy of the multi-joint robots. Since the initial (t = 0) and the terminal (t = T ) conditions are fixed, we can introduce a scalar constant c = E(T ) − E(0). Then, the energy relationship can be rewritten due to the ˙ orthogonality of τc to q.
6.2 Analysis on torque minimization problem
∫
Here, we analytically derive that the optimal actuator torque τ opt , which minimizes the cost function J, can be described as linear state feedback of the angular velocity ˙ q. At first, we decompose the actuator torque τ to a component proportional to the angular velocity q˙ and a remain˙ Then, ing component τc = (τc1 , τc2 , · · · τcn )T = τ − Aq. the actuator torque can be described by τ = Aq˙ + τc . The coefficients ai (i = 1, 2, · · · n) of the component of the ∫ T
τi q˙i dt
angular velocity are calculated by ai = ∫0 T 0
q˙i2 dt
. There-
0.001
τ1 (b) a1
0.000
295
300
Time[s]
Torque [Nm]
(a)
Torque [Nm]
fore, the remaining component τci does not include the component of the angular velocity q˙i , and inner product ∫T between the two components 0 q˙i τci dt will be 0, because the two components have a orthogonal relationship. 0.04
τ2 a2
0.00 -0.04
T
q˙ τ dt = T
0
∫
T 0
˙ 0,T ˙ 0,T q˙ T (Aq˙ + τc )dt = ||q|| A = c + ||q|| D (19)
Based on the above energy analysis, we derive the optimal actuator torque τ opt as follows. First, we substitute the decomposed components of the actuator torque τ = Aq˙ + τc into the cost function J. J=
∫
T 0
{
} ′ ′ q˙ T (A − D)q˙ + 2q˙ T AT A τc + τc T A τc dt (20)
By using the orthogonality of τci to q˙i and the energy relationship of the equation (19), we obtain J =c+
∫
T
′
τc T A τc dt.
(21)
0
Obviously, the cost function J will be the smallest when τc = 0. Therefore, the optimal actuator torque τ opt can ˙ be described by the linear state feedback form τ opt = Aq. 6.3 Discussion
295
300
Time[s]
Fig. 5. Converged Actuator Torque of Case 2
Physical meaning of the above analysis is that minimum actuator torque is required to generate motions without
exerting torque, which does not contribute the energy supply. 6.4 Analysis Based on Hamilton-Jacobi-Bellman Equation We can lead to similar conclusion τ opt = Aq˙ by using the Hamilton-Jacobi-Bellman equation. This kind of analysis by using Hamilton-Jacobi-Bellman equation is proposed by Arimoto et al.(10). However, Arimoto et al.considered only a damping shaping problem of position control, and they did not consider problems of motion generation. In this analysis, we ignore the viscosity D = 0, and we ∫T set the cost function as J = 0 τ A 1 τ dt instead of the equation (18). Then, the Hamiltonian [ { is given by H} = ˙ ˙ q˙ + S(q, q) τ T A 1 τ + λ1 T p + λ2 T R(q) 1 − 12 R(q) ˙ −g(q) − K(q − q e ) + τ ], where p = q. We can obtain the conclusion τ opt = Aq˙ by using the analysis of the paper (10). A difference from the paper (10) is to set the value function v to −E(t). Therefore, it is proved by using the Hamilton-Jacobi-Bellman equation that the optimal control can be described by linear state feedback. 6.5 Comparison of Proposed Formulation with Conventional Resonance Next, we discuss analogies and differences between conventional resonance and the discussion in this paper. Conventional resonance is formulated for 1-DOF linear dynamics m¨ q + dq˙ + kq = τ where m ∈ ℜ is an inertia coefficient, d ∈ ℜ is a viscosity, k ∈ ℜ is stiffness, q ∈ ℜ is displacement, and τ ∈ ℜ is actuator torque. In the resonant condition, motions will be sinusoidal q˙ = ad sin(ωt + ϕ), and stiffness k satisfies the condition k = mω 2 . Then, the dynamics will be equivalent to a pure viscous system dq˙ = τ , and the actuator torque will be linear state feedback of the velocity q. ˙ In the resonant condition, the periodic motions q(t) = q(t + T ) ∫T are generated by minimum actuator torque 0 τ 2 dt. In our discussion, the actuator torque is linear state ˙ Therefore, the feedback of the angular velocity τ = Aq. property of linear state feedback is the same between the proposed formulation and conventional resonance. Generating motions by minimum actuator torque is also the same. Therefore, our proposed formulation can be regarded as a kind of extension of resonance to the periodic motions of the multi-joint robots. 7. CONCLUSION This paper proposes a new controller to realize an effective simultaneous use of passive elements and actuators . This proposed controller is composed of adaptive parameter tunings, delayed feedback control, and linear state feedback of angular velocity. Advantages of our proposed controller are to work without using exact parameter values of the controlled system nor huge numerical calculation. We analyzed that the closed loop systems are locally stable.
Simulation results demonstrated the effectiveness of the proposed controller and proposed resonance. Additionally, we discuss an energy saving effect of linear state feedback of angular velocity. Then, a minimum torque controller can be represented as linear state feed˙ Physical meaning of this property is that back τ = Aq. minimum torque is required to generate periodic motions without exerting torque, which does not contribute energy supply. This property is the same as conventional resonance. To prove global stability of the controlled systems with the proposed controller is our important future work. 8. ACKNOWLEDGMENT This work has been partially supported by a Grantin-Aid for Young Scientists (Start-up) ”An Extension of Resonance to Multi-Joint Structures and Adaptive Control Methods Based on Extended Resonance” from the Japanese Ministry of Education, Culture, Sports, Science and Technology. REFERENCES [1] S. Satoh, K. Fujimoto, S. Hyon: “Gait generation for passive running via iterative learning control”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2006, pp. 5907-5912, 2008. [2] S. Hyon, T. Emura: “Symmetric walking control: Invariance and global stability”, IEEE International Conference on Robotics and Automation 2005, pp. 1455-1462, 2005. [3] M. Uemura, S. Kawamura: “An Energy Saving Control Method of Robot Motions based on Adaptive Stiffness Optimization - Cases of Multi-Frequency Components -”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2008, pp. 551-557, 2008. [4] M. Uemura, G. Lu, S. Kawamura, S. Ma: “Passive Periodic Motions of Multi-Joint Robots by Stiffness Adaptation and DFC for Energy Saving”, SICE Annual Conference 2008, pp. 2853-2858, 2008. [5] M. Uemura, S. Kawamura: “Resonance-based Motion Control Method for Multi-Joint Robot through Combining Stiffness Adaptation and Iterative Learning Control”, IEEE International Conference on Robotics and Automation 2009, pp. 1543-1548, 2009. [6] K. Pyragas: “Continuous Control of Chaos by SelfControlling Feedback”, Physics Letters A, Vol. 170, No. 6, pp. 412-428, 1992. [7] S. Collins, A. Ruina, R. Tedrake, M. Wisse: “Efficient bipedal robots based on passive-dynamic walkers,” Science, vol. 307, pp. 1082-1085, 2005. [8] F. Asano, Z.W. Luo, M. Yamakita: “Unification of Dynamic Gait Generation Methods via Variable Virtual Gravity and Its Control Performance Analysis”, IEEE/RSJ International Conference on Intelligent Robots and Systems 2004, pp. 3865-3870, 2004. [9] S. Arimoto: Control Theory of Non-Linear Mechanical Systems: A Passivity-Based and Circuit-Theoretic Approach. Oxford Engineering Science Series, 1996. [10] S. Arimoto: “Extension of Impedance Matching to Nonlinear Dynamics of Robotic Tasks”, System and Control Letters, Vol. 36, No. 2, pp. 109-119, 1998.