Energy Efficiency of Quadruped Gaits - Semantic Scholar
Energy Efficiency of Quadruped Gaits M. F. Silva and J. A. Tenreiro Machado Department of Electrical Engineering, Institute of Engineering of Porto, Portugal [email protected]
Abstract This paper studies periodic gaits of quadruped locomotion systems. The purpose is to determine the best set of gait and locomotion variables during walking, for different robot velocities, based on two formulated performance measures. A set of experiments reveals the influence of the gait and locomotion variables upon the proposed indices, namely that the gait and the locomotion parameters should be adapted to the robot forward velocity. Keywords: Robotics, locomotion, modelling, simulation.
1 Introduction Walking machines allow locomotion in terrain inaccessible to other type of vehicles. For this to become possible, gait analysis and selection is a research area requiring an appreciable modelling effort for the improvement of mobility with legs in unstructured environments. Several robots have been developed which adopt different quadruped gaits such as the bound [1 – 3], trot [4] and gallop [5]. Nevertheless, detailed studies on the best set of gait and locomotion variables for different robot velocities are missing [6]. In this line of thought, a simulation model for multilegged locomotion systems was developed, for several periodic gaits [7]. Based on this model, we test the quadruped robot locomotion, as a function of VF, when adopting different gaits often observed in several quadruped animals while they walk / run at variable speeds [8]. This study intends to generalize previous work [9] through the formulation of two indices measuring the average energy consumption and the hip trajectory errors during forward straight line walking at different velocities.
736 M. F. Silva and J. A. Tenreiro Machado VF
Body Compliance
Trajectory Planning
21 41 SP
L41
pHd(t)
pFd(t)
22
HB
y
42 O4
x
FRF(t)
JT F τF(t)
θd(t) +
eθ(t)
L22 L42
Foot force feedback
Inverse Kinematic Model
L21
G c1(s)
τθ(t) +
−
Environment
actuators eτ(t)
G c2
τC(t)
τm(t)
Robot
−
O2
θ(t)
(x2F,y2F)
FC LS
Position / velocity feedback
Ground Compliance
Fig. 1. Quadruped robot model and control architecture
Bearing these facts in mind, the paper is organized as follows. Section two introduces the robot model and control architecture and section three presents the optimizing indices. Section four develops a set of experiments that reveal the influence of the locomotion parameters and robot gaits on the performance measures, as a function of robot body velocity. Finally, section five outlines the main conclusions.
2 Quadruped robot model and control architecture We consider a quadruped walking system (Fig. 1) with n = 4 legs, equally distributed along both sides of the robot body, having each two rotational joints (i.e., j = {1, 2} ≡ {hip, knee}). Motion is described by means of a world coordinate system. The kinematic model comprises: the cycle time T, the duty factor β, the transference time tT = (1−β)T, the support time tS = βT, the step length LS, the stroke pitch SP, the body height HB, the maximum foot clearance FC, the ith leg lengths Li1 and Li2 and the foot trajectory offset Oi (i = 1, …, n). The forward motion planning algorithm accepts, as inputs, the desired cartesian trajectories of the legs hips pHd(t) = [xiHd(t), yiHd(t)]T (horizontal movement with a constant forward speed VF = LS / T) and feet pFd(t) = [xiFd(t), yiFd(t)]T (periodic trajectory for each foot, being the trajectory of the swing leg foot computed through a cycloid function [7]) and, by means of an inverse kinematics algorithm, generates the related joint trajectories Θd(t) = [θi1d(t), θi2d(t)]T, selecting the solution corresponding to a forward knee [7]. Concerning the dynamic model, the robot body is divided in n identical segments (each with mass Mbn−1) and a linear spring-damper system is adopted to implement the intra-body compliance (Fig. 1) [7], being its parameters defined so that the body behaviour is similar to the one expected
Energy Efficiency of Quadruped Gaits
737
to occur on an animal (Table 1). The contact of the ith robot foot with the ground is modelled through a non-linear system [7] with linear stiffness and non-linear damping (Fig. 1). The values for the parameters are based on the studies of soil mechanics (Table 1). The robot inverse dynamic model is formulated as:
(
)
&& + c Θ,Θ & + g ( Θ ) − F − J T (Θ)F Γ = H (Θ) Θ RH RF
(1)
where Γ = [fix, fiy, τi1, τi2]T (i = 1, …, n) is the vector of forces / torques, Θ = [xiH, yiH, θi1, θi2]T is the vector of position coordinates, H(Θ) is the & and g(Θ) are the vectors of centrifuinertia matrix and c Θ,Θ
(
)
gal / Coriolis and gravitational forces / torques, respectively. The (m+2) × 2 (m = 2) matrix JT(Θ) is the transpose of the robot Jacobian matrix, FRH is the (m+2) × 1 vector of the body inter-segment forces and FRF is the 2 × 1 vector of the reaction forces that the ground exerts on the robot feet. These forces are null during the foot transfer phase. Furthermore, we consider that the joint actuators are not ideal, exhibiting saturation, being τijC the controller demanded torque, τijMax the maximum torque that the actuator can supply and τijm the motor effective torque. The general control architecture of the multi-legged locomotion system is presented in Fig. 1. The control algorithm considers an external position and velocity feedback and an internal feedback loop with information of foot-ground interaction force [10]. For Gc1(s) we adopt a PD controller and for Gc2 a simple P controller. For the PD algorithm we have:
GC1 j ( s ) = Kp j + Kd j s,
j = 1, 2
(2)
being Kpj and Kdj the proportional and derivative gains. Table 1. System parameters Robot model parameters SP 1m Lij 0.5 m Oi 0m Mb 88.0 kg Mij 1 kg KxH 105 Nm−1 KyH 104 Nm−1 BxH 103 Nsm−1 ByH 102 Nsm−1
Locomotion parameters LS 1m HB 0.9 m FC 0.1 m Ground parameters KxF 1302152.0 Nm−1 KyF 1705199.0 Nm−1 BxF 2364932.0 Nsm−1 ByF 2706233.0 Nsm−1
738 M. F. Silva and J. A. Tenreiro Machado
3 Measures for performance evaluation Two global measures of the overall mechanism performance in an average sense are established [9]. The first, the mean absolute density of energy per travelled distance (Eav), is obtained by averaging the mechanical absolute energy delivered over the travelled distance d: Eav =
1 n m T ∑∑ τ ijm ( t ) θ& ij ( t ) dt d i =1 j =1 ∫0
Jm −1
(3)
The other, based on the hip trajectory tracking errors (εxyH), is defined as: n
ε xyH = ∑ i =1
1 NS
NS
∑(∆ k =1
2 ixH
+ ∆ iyH 2 ) , ∆ iη H = ηiHd (k ) − ηiH (k ), η = { x, y}
[ m] (4)
where Ns is the total number of samples for averaging purposes. The performance optimization requires the minimization of each index.
4 Simulation results To illustrate the use of the preceding concepts, in this section we develop a set of simulation experiments to estimate the influence of parameters LS and HB, when adopting periodic gaits [8]. We consider three walking gaits (Walk, Chelonian Walk and Amble), two symmetrical running gaits (Trot and Pace) and five asymmetrical running gaits (Canter, Transverse Gallop, Rotary Gallop, Half-Bound and Bound). These gaits are usually adopted by animals moving at low, moderate and high speed, respectively. In a first phase, the robot is simulated in order to analyse the evolution of the locomotion parameters LS and HB with VF, being the controller tuned for each gait while the robot is walking with VF = 1 ms−1. In a second phase, the controller is tuned for each particular gait and locomotion velocity and the quadruped robot is then simulated in order to compare the performance of the different gaits versus VF. For the system simulation we consider the robot body parameters, the locomotion parameters and the ground parameters presented in Table 1. Moreover, we assume high performance joint actuators with a maximum torque of τijMax = 400 Nm. To tune the controller we adopt a systematic method, testing and evaluating a grid of several possible combinations of controller parameters, while minimising Eav (3).
Energy Efficiency of Quadruped Gaits
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Table 2. Quadruped controller parameters Gait Walk Chelonian Walk Amble Trot Pace Canter Transverse Gallop Rotary Gallop Half-Bound Bound
φ1
φ2
0 0 0 0 0 0 0 0 0.7 0
0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.1 0.6 0
φ3
φ4
β
Kp1 0.75 0.25 0.65 1000 0.5 0 0.8 5000 0.75 0.25 0.45 1000 0.5 0 0.4 1000 0 0.5 0.4 1000 0.7 0 0.4 1000 0.6 0.8 0.3 6000 0.6 0.5 0.3 4000 0 0 0.2 4000 0.5 0.5 0.2 2000
Kd1 40 200 20 140 60 0 40 0 0 0
Kp2 2000 2500 1000 2000 500 1500 1000 500 3000 500
Kd2 40 20 60 20 40 20 40 80 20 20
4.1 Locomotion parameters versus body forward velocity
In order to analyse the evolution of the locomotion parameters LS and HB with VF, we test the forward straight line quadruped robot locomotion, as a function of VF, when adopting different gaits often observed in several quadruped animals while they walk / run at variable speeds [8]. With this purpose, the robot controller is tuned for each gait, considering the forward velocity VF = 1.0 ms−1, resulting the possible controller parameters presented in Table 2. After completing the controller tuning, the robot forward straight line locomotion is simulated for different gaits, while varying the body velocity on the range 0.2 ≤ VF ≤ 5.0 ms−1. For each gait and body velocity, the set of locomotion parameters (LS, HB) that minimises the performance index Eav is determined. The chart presented in Fig. 2 depicts the minimum value of the index Eav, on the range of VF under consideration, for three different robot gaits. It is possible to conclude that the minimum values of the index Eav increase with VF, independently of the adopted locomotion gait. Although not presented here, due to space limitations, the behaviour of the charts min[Eav(VF)], for all other gaits present similar shapes. Next we analyse how the locomotion parameters vary with VF. Figure 3 (left) shows, for three locomotion gaits, that the optimal value of LS must increase with VF when considering the performance index Eav. The same figure (right) shows that HB must decrease with VF from the viewpoint of the same performance index. For the other periodic walking gaits considered on this study, the evolution of the optimization index Eav and the locomotion parameters (LS, HB) with VF follows the same pattern.
740 M. F. Silva and J. A. Tenreiro Machado 2500
Walk Pace Bound
min(Eav ) (Jm−1)
2000
1500
1000
500
0 0.0
1.0
2.0
3.0
4.0
5.0
V F (ms −1)
Fig. 2. min[Eav(VF)] for FC = 0.1 m 2.5
1.0 0.9 0.8
2.0
HB (m)
LS (m)
0.7
1.5
1.0
0.6 0.5 0.4 0.3
0.5
Amble Pace Transverse Gallop
0.0 0.0
1.0
2.0
3.0
4.0
5.0
0.2
Walk Chelonian Walk Amble
0.1 0.0 0.0
1.0
2.0
V F (ms−1)
3.0
4.0
5.0
V F (ms−1)
Fig. 3. LS(VF) (left) and HB(VF) (right) for min(Eav), with FC = 0.1 m
Therefore, we conclude that the locomotion parameters should be adapted to the walking velocity in order to optimize the robot performance. As VF increases, the value of HB should be decreased and the value of LS increased. These results seem to agree with the observations of the living quadruped creatures [11]. 4.2 Gait selection versus body forward velocity
In a second phase we determine the best locomotion gait, from the viewpoint of energy efficiency, at each forward robot velocity on the range 0.1 ≤ VF ≤ 10.0 ms−1. For this phase of the study, the controller is tuned for each particular locomotion velocity, while minimizing the index Eav, and adopting the locomotion parameters LS = 1.0 m and HB = 0.9 m. Figure 4 presents the charts of min[Eav(VF)] and the corresponding value of εxyH for the different gaits. The index Eav suggests that the locomotion should be Amble, Bound and Half-Bound as the speed increases. The other gaits under consideration present values of min[Eav(VF)] higher than these ones, on all range of VF under consideration. In particular, the gaits Walk and Chelonian Walk present the higher values of this performance measure.
Energy Efficiency of Quadruped Gaits 1000.0
741
ε xyH (m)
min(Eav ) (Jm−1)
10.0
Amble Half-Bound Bound
100.0 0.1
1.0 V F (ms -1)
10.0
1.0
Walk
Chelonian Walk
Amble
Transverse Gallop
0.1 0.1
1.0
10.0
V F (ms −1)
Fig. 4. min[Eav(VF)] and the corresponding value of εxyH, for FC = 0.1 m
Analysing the locomotion though the index εxyH, we verify that for low values of VF (VF < 1ms−1), the gaits Walk and Chelonian Walk allow the lower oscillations of the hips. For increasing values of the locomotion velocity the Amble and Transverse Gallop gaits present the lower values of εxyH. From these results, we can conclude that, from the viewpoint of each proposed optimising index, the robot gait should change with the desired forward body velocity. These results seem to agree with the observations of the living quadruped creatures [11]. In conclusion, the locomotion gait and the parameters LS and HB should be chosen according to the intended robot forward velocity in order to optimize the energy efficiency or the oscillation of the hips trajectories.
5 Conclusions In this paper we have compared several aspects of periodic quadruped locomotion gaits. By implementing different motion patterns, we estimated how the robot responds to the locomotion parameters step length and body height and to the forward speed. For analyzing the system performance two quantitative measures were defined based on the system energy consumption and the trajectory errors. A set of experiments determined the best set of gait and locomotion variables, as a function of the forward velocity VF. The results show that the locomotion parameters should be adapted to the walking velocity in order to optimize the robot performance. As the forward velocity increases, the value of HB should be decreased and the value of LS increased. Furthermore, for the case of a quadruped robot, we concluded that the gait should be adapted to VF. While our focus has been on a dynamic analysis in periodic gaits, certain aspects of locomotion are not necessarily captured by the proposed
742 M. F. Silva and J. A. Tenreiro Machado
measures. Consequently, future work in this area will address the refinement of our models to incorporate more unstructured terrains, namely with distinct trajectory planning concepts. The effect of distinct values of the robot intra-body compliance parameters will also be studied, since animals use their body compliance to store energy at high velocities.
References 1.
Poulakakis I., Smith J. A. and Buehler M. (2004) Experimentally Validated Bounding Models for the Scout II Quadruped Robot. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 2. Zhang Z. G., Fukuoka Y. and Kimura H. (2004) Stable Quadrupedal Running Based on a Spring-Loaded Two-Segmented Legged Model. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 3. Iida F. and Pfeifer R. (2004) ”Cheap” Rapid Locomotion of a Quadruped Robot: Self-Stabilization of Bounding Gait. Proc. of the 8th Conf. on Intelligent Autonomous Systems (IAS’2004), Amsterdam, The Netherlands, 10-13 March. 4. Kohl N. and Stone P. (2004) Policy Gradient Reinforcement Learning for Fast Quadrupedal Locomotion. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 5. Palmer L. R., Orin D. E., Marhefka D. W., Schmiedeler J. P. and Waldron K. J. (2003) Intelligent Control of an Experimental Articulated Leg for a Galloping Machine. Proc. of the 2003 IEEE Int. Conf. on Rob. & Aut. (ICRA’2003), Taipei, Taiwan, 14-19 September. 6. Hardt M. and von Stryk O (2000) Towards Optimal Hybrid Control Solutions for Gait Patterns of a Quadruped. Proc. 3rd Int. Conf. on Climbing and Walking Robots (CLAWAR 2000), Madrid, Spain, 2 – 4 October. 7. Silva M. F., Machado J. A. T. and Lopes A. M. (2005) Modeling and Simulation of Artificial Locomotion Systems. ROBOTICA (accepted for publication). 8. URL: http://www.biology.leeds.ac.uk/teaching/3rdyear/Blgy3120/Jmvr/ /Loco/Gaits/GAITS.htm [2005-06-30] 9. Silva M. F., Machado J. A. T., Lopes A. M. and Tar J. K. (2004) Gait Selection for Quadruped and Hexapod Walking Systems. Proc. of the 2004 IEEE Int. Conf. on Computational Cybernetics (ICCC’2004), Vienna, Austria, 30 August – 1 September. 10. Silva M. F., Machado J. A. T. and Lopes A. M. (2003) Position / Force Control of a Walking Robot. Machine Intelligence and Robotic Control, vol. 5, no. 2, pp. 33–44. 11. Alexander R. McN. (1984) The Gaits of Bipedal and Quadrupedal Animal. The Int. J. of Robotics Research, vol. 3, no. 2, pp. 49–59.
Abstract This paper studies periodic gaits of quadruped locomotion systems. The purpose is to determine the best set of gait and locomotion variables during walking, for different robot velocities, based on two formulated performance measures. A set of experiments reveals the influence of the gait and locomotion variables upon the proposed indices, namely that the gait and the locomotion parameters should be adapted to the robot forward velocity. Keywords: Robotics, locomotion, modelling, simulation.
1 Introduction Walking machines allow locomotion in terrain inaccessible to other type of vehicles. For this to become possible, gait analysis and selection is a research area requiring an appreciable modelling effort for the improvement of mobility with legs in unstructured environments. Several robots have been developed which adopt different quadruped gaits such as the bound [1 – 3], trot [4] and gallop [5]. Nevertheless, detailed studies on the best set of gait and locomotion variables for different robot velocities are missing [6]. In this line of thought, a simulation model for multilegged locomotion systems was developed, for several periodic gaits [7]. Based on this model, we test the quadruped robot locomotion, as a function of VF, when adopting different gaits often observed in several quadruped animals while they walk / run at variable speeds [8]. This study intends to generalize previous work [9] through the formulation of two indices measuring the average energy consumption and the hip trajectory errors during forward straight line walking at different velocities.
736 M. F. Silva and J. A. Tenreiro Machado VF
Body Compliance
Trajectory Planning
21 41 SP
L41
pHd(t)
pFd(t)
22
HB
y
42 O4
x
FRF(t)
JT F τF(t)
θd(t) +
eθ(t)
L22 L42
Foot force feedback
Inverse Kinematic Model
L21
G c1(s)
τθ(t) +
−
Environment
actuators eτ(t)
G c2
τC(t)
τm(t)
Robot
−
O2
θ(t)
(x2F,y2F)
FC LS
Position / velocity feedback
Ground Compliance
Fig. 1. Quadruped robot model and control architecture
Bearing these facts in mind, the paper is organized as follows. Section two introduces the robot model and control architecture and section three presents the optimizing indices. Section four develops a set of experiments that reveal the influence of the locomotion parameters and robot gaits on the performance measures, as a function of robot body velocity. Finally, section five outlines the main conclusions.
2 Quadruped robot model and control architecture We consider a quadruped walking system (Fig. 1) with n = 4 legs, equally distributed along both sides of the robot body, having each two rotational joints (i.e., j = {1, 2} ≡ {hip, knee}). Motion is described by means of a world coordinate system. The kinematic model comprises: the cycle time T, the duty factor β, the transference time tT = (1−β)T, the support time tS = βT, the step length LS, the stroke pitch SP, the body height HB, the maximum foot clearance FC, the ith leg lengths Li1 and Li2 and the foot trajectory offset Oi (i = 1, …, n). The forward motion planning algorithm accepts, as inputs, the desired cartesian trajectories of the legs hips pHd(t) = [xiHd(t), yiHd(t)]T (horizontal movement with a constant forward speed VF = LS / T) and feet pFd(t) = [xiFd(t), yiFd(t)]T (periodic trajectory for each foot, being the trajectory of the swing leg foot computed through a cycloid function [7]) and, by means of an inverse kinematics algorithm, generates the related joint trajectories Θd(t) = [θi1d(t), θi2d(t)]T, selecting the solution corresponding to a forward knee [7]. Concerning the dynamic model, the robot body is divided in n identical segments (each with mass Mbn−1) and a linear spring-damper system is adopted to implement the intra-body compliance (Fig. 1) [7], being its parameters defined so that the body behaviour is similar to the one expected
Energy Efficiency of Quadruped Gaits
737
to occur on an animal (Table 1). The contact of the ith robot foot with the ground is modelled through a non-linear system [7] with linear stiffness and non-linear damping (Fig. 1). The values for the parameters are based on the studies of soil mechanics (Table 1). The robot inverse dynamic model is formulated as:
(
)
&& + c Θ,Θ & + g ( Θ ) − F − J T (Θ)F Γ = H (Θ) Θ RH RF
(1)
where Γ = [fix, fiy, τi1, τi2]T (i = 1, …, n) is the vector of forces / torques, Θ = [xiH, yiH, θi1, θi2]T is the vector of position coordinates, H(Θ) is the & and g(Θ) are the vectors of centrifuinertia matrix and c Θ,Θ
(
)
gal / Coriolis and gravitational forces / torques, respectively. The (m+2) × 2 (m = 2) matrix JT(Θ) is the transpose of the robot Jacobian matrix, FRH is the (m+2) × 1 vector of the body inter-segment forces and FRF is the 2 × 1 vector of the reaction forces that the ground exerts on the robot feet. These forces are null during the foot transfer phase. Furthermore, we consider that the joint actuators are not ideal, exhibiting saturation, being τijC the controller demanded torque, τijMax the maximum torque that the actuator can supply and τijm the motor effective torque. The general control architecture of the multi-legged locomotion system is presented in Fig. 1. The control algorithm considers an external position and velocity feedback and an internal feedback loop with information of foot-ground interaction force [10]. For Gc1(s) we adopt a PD controller and for Gc2 a simple P controller. For the PD algorithm we have:
GC1 j ( s ) = Kp j + Kd j s,
j = 1, 2
(2)
being Kpj and Kdj the proportional and derivative gains. Table 1. System parameters Robot model parameters SP 1m Lij 0.5 m Oi 0m Mb 88.0 kg Mij 1 kg KxH 105 Nm−1 KyH 104 Nm−1 BxH 103 Nsm−1 ByH 102 Nsm−1
Locomotion parameters LS 1m HB 0.9 m FC 0.1 m Ground parameters KxF 1302152.0 Nm−1 KyF 1705199.0 Nm−1 BxF 2364932.0 Nsm−1 ByF 2706233.0 Nsm−1
738 M. F. Silva and J. A. Tenreiro Machado
3 Measures for performance evaluation Two global measures of the overall mechanism performance in an average sense are established [9]. The first, the mean absolute density of energy per travelled distance (Eav), is obtained by averaging the mechanical absolute energy delivered over the travelled distance d: Eav =
1 n m T ∑∑ τ ijm ( t ) θ& ij ( t ) dt d i =1 j =1 ∫0
Jm −1
(3)
The other, based on the hip trajectory tracking errors (εxyH), is defined as: n
ε xyH = ∑ i =1
1 NS
NS
∑(∆ k =1
2 ixH
+ ∆ iyH 2 ) , ∆ iη H = ηiHd (k ) − ηiH (k ), η = { x, y}
[ m] (4)
where Ns is the total number of samples for averaging purposes. The performance optimization requires the minimization of each index.
4 Simulation results To illustrate the use of the preceding concepts, in this section we develop a set of simulation experiments to estimate the influence of parameters LS and HB, when adopting periodic gaits [8]. We consider three walking gaits (Walk, Chelonian Walk and Amble), two symmetrical running gaits (Trot and Pace) and five asymmetrical running gaits (Canter, Transverse Gallop, Rotary Gallop, Half-Bound and Bound). These gaits are usually adopted by animals moving at low, moderate and high speed, respectively. In a first phase, the robot is simulated in order to analyse the evolution of the locomotion parameters LS and HB with VF, being the controller tuned for each gait while the robot is walking with VF = 1 ms−1. In a second phase, the controller is tuned for each particular gait and locomotion velocity and the quadruped robot is then simulated in order to compare the performance of the different gaits versus VF. For the system simulation we consider the robot body parameters, the locomotion parameters and the ground parameters presented in Table 1. Moreover, we assume high performance joint actuators with a maximum torque of τijMax = 400 Nm. To tune the controller we adopt a systematic method, testing and evaluating a grid of several possible combinations of controller parameters, while minimising Eav (3).
Energy Efficiency of Quadruped Gaits
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Table 2. Quadruped controller parameters Gait Walk Chelonian Walk Amble Trot Pace Canter Transverse Gallop Rotary Gallop Half-Bound Bound
φ1
φ2
0 0 0 0 0 0 0 0 0.7 0
0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.1 0.6 0
φ3
φ4
β
Kp1 0.75 0.25 0.65 1000 0.5 0 0.8 5000 0.75 0.25 0.45 1000 0.5 0 0.4 1000 0 0.5 0.4 1000 0.7 0 0.4 1000 0.6 0.8 0.3 6000 0.6 0.5 0.3 4000 0 0 0.2 4000 0.5 0.5 0.2 2000
Kd1 40 200 20 140 60 0 40 0 0 0
Kp2 2000 2500 1000 2000 500 1500 1000 500 3000 500
Kd2 40 20 60 20 40 20 40 80 20 20
4.1 Locomotion parameters versus body forward velocity
In order to analyse the evolution of the locomotion parameters LS and HB with VF, we test the forward straight line quadruped robot locomotion, as a function of VF, when adopting different gaits often observed in several quadruped animals while they walk / run at variable speeds [8]. With this purpose, the robot controller is tuned for each gait, considering the forward velocity VF = 1.0 ms−1, resulting the possible controller parameters presented in Table 2. After completing the controller tuning, the robot forward straight line locomotion is simulated for different gaits, while varying the body velocity on the range 0.2 ≤ VF ≤ 5.0 ms−1. For each gait and body velocity, the set of locomotion parameters (LS, HB) that minimises the performance index Eav is determined. The chart presented in Fig. 2 depicts the minimum value of the index Eav, on the range of VF under consideration, for three different robot gaits. It is possible to conclude that the minimum values of the index Eav increase with VF, independently of the adopted locomotion gait. Although not presented here, due to space limitations, the behaviour of the charts min[Eav(VF)], for all other gaits present similar shapes. Next we analyse how the locomotion parameters vary with VF. Figure 3 (left) shows, for three locomotion gaits, that the optimal value of LS must increase with VF when considering the performance index Eav. The same figure (right) shows that HB must decrease with VF from the viewpoint of the same performance index. For the other periodic walking gaits considered on this study, the evolution of the optimization index Eav and the locomotion parameters (LS, HB) with VF follows the same pattern.
740 M. F. Silva and J. A. Tenreiro Machado 2500
Walk Pace Bound
min(Eav ) (Jm−1)
2000
1500
1000
500
0 0.0
1.0
2.0
3.0
4.0
5.0
V F (ms −1)
Fig. 2. min[Eav(VF)] for FC = 0.1 m 2.5
1.0 0.9 0.8
2.0
HB (m)
LS (m)
0.7
1.5
1.0
0.6 0.5 0.4 0.3
0.5
Amble Pace Transverse Gallop
0.0 0.0
1.0
2.0
3.0
4.0
5.0
0.2
Walk Chelonian Walk Amble
0.1 0.0 0.0
1.0
2.0
V F (ms−1)
3.0
4.0
5.0
V F (ms−1)
Fig. 3. LS(VF) (left) and HB(VF) (right) for min(Eav), with FC = 0.1 m
Therefore, we conclude that the locomotion parameters should be adapted to the walking velocity in order to optimize the robot performance. As VF increases, the value of HB should be decreased and the value of LS increased. These results seem to agree with the observations of the living quadruped creatures [11]. 4.2 Gait selection versus body forward velocity
In a second phase we determine the best locomotion gait, from the viewpoint of energy efficiency, at each forward robot velocity on the range 0.1 ≤ VF ≤ 10.0 ms−1. For this phase of the study, the controller is tuned for each particular locomotion velocity, while minimizing the index Eav, and adopting the locomotion parameters LS = 1.0 m and HB = 0.9 m. Figure 4 presents the charts of min[Eav(VF)] and the corresponding value of εxyH for the different gaits. The index Eav suggests that the locomotion should be Amble, Bound and Half-Bound as the speed increases. The other gaits under consideration present values of min[Eav(VF)] higher than these ones, on all range of VF under consideration. In particular, the gaits Walk and Chelonian Walk present the higher values of this performance measure.
Energy Efficiency of Quadruped Gaits 1000.0
741
ε xyH (m)
min(Eav ) (Jm−1)
10.0
Amble Half-Bound Bound
100.0 0.1
1.0 V F (ms -1)
10.0
1.0
Walk
Chelonian Walk
Amble
Transverse Gallop
0.1 0.1
1.0
10.0
V F (ms −1)
Fig. 4. min[Eav(VF)] and the corresponding value of εxyH, for FC = 0.1 m
Analysing the locomotion though the index εxyH, we verify that for low values of VF (VF < 1ms−1), the gaits Walk and Chelonian Walk allow the lower oscillations of the hips. For increasing values of the locomotion velocity the Amble and Transverse Gallop gaits present the lower values of εxyH. From these results, we can conclude that, from the viewpoint of each proposed optimising index, the robot gait should change with the desired forward body velocity. These results seem to agree with the observations of the living quadruped creatures [11]. In conclusion, the locomotion gait and the parameters LS and HB should be chosen according to the intended robot forward velocity in order to optimize the energy efficiency or the oscillation of the hips trajectories.
5 Conclusions In this paper we have compared several aspects of periodic quadruped locomotion gaits. By implementing different motion patterns, we estimated how the robot responds to the locomotion parameters step length and body height and to the forward speed. For analyzing the system performance two quantitative measures were defined based on the system energy consumption and the trajectory errors. A set of experiments determined the best set of gait and locomotion variables, as a function of the forward velocity VF. The results show that the locomotion parameters should be adapted to the walking velocity in order to optimize the robot performance. As the forward velocity increases, the value of HB should be decreased and the value of LS increased. Furthermore, for the case of a quadruped robot, we concluded that the gait should be adapted to VF. While our focus has been on a dynamic analysis in periodic gaits, certain aspects of locomotion are not necessarily captured by the proposed
742 M. F. Silva and J. A. Tenreiro Machado
measures. Consequently, future work in this area will address the refinement of our models to incorporate more unstructured terrains, namely with distinct trajectory planning concepts. The effect of distinct values of the robot intra-body compliance parameters will also be studied, since animals use their body compliance to store energy at high velocities.
References 1.
Poulakakis I., Smith J. A. and Buehler M. (2004) Experimentally Validated Bounding Models for the Scout II Quadruped Robot. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 2. Zhang Z. G., Fukuoka Y. and Kimura H. (2004) Stable Quadrupedal Running Based on a Spring-Loaded Two-Segmented Legged Model. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 3. Iida F. and Pfeifer R. (2004) ”Cheap” Rapid Locomotion of a Quadruped Robot: Self-Stabilization of Bounding Gait. Proc. of the 8th Conf. on Intelligent Autonomous Systems (IAS’2004), Amsterdam, The Netherlands, 10-13 March. 4. Kohl N. and Stone P. (2004) Policy Gradient Reinforcement Learning for Fast Quadrupedal Locomotion. Proc. of the 2004 IEEE Int. Conf. on Rob. & Aut. (ICRA’2004), New Orleans, USA, 26 April – 1 May. 5. Palmer L. R., Orin D. E., Marhefka D. W., Schmiedeler J. P. and Waldron K. J. (2003) Intelligent Control of an Experimental Articulated Leg for a Galloping Machine. Proc. of the 2003 IEEE Int. Conf. on Rob. & Aut. (ICRA’2003), Taipei, Taiwan, 14-19 September. 6. Hardt M. and von Stryk O (2000) Towards Optimal Hybrid Control Solutions for Gait Patterns of a Quadruped. Proc. 3rd Int. Conf. on Climbing and Walking Robots (CLAWAR 2000), Madrid, Spain, 2 – 4 October. 7. Silva M. F., Machado J. A. T. and Lopes A. M. (2005) Modeling and Simulation of Artificial Locomotion Systems. ROBOTICA (accepted for publication). 8. URL: http://www.biology.leeds.ac.uk/teaching/3rdyear/Blgy3120/Jmvr/ /Loco/Gaits/GAITS.htm [2005-06-30] 9. Silva M. F., Machado J. A. T., Lopes A. M. and Tar J. K. (2004) Gait Selection for Quadruped and Hexapod Walking Systems. Proc. of the 2004 IEEE Int. Conf. on Computational Cybernetics (ICCC’2004), Vienna, Austria, 30 August – 1 September. 10. Silva M. F., Machado J. A. T. and Lopes A. M. (2003) Position / Force Control of a Walking Robot. Machine Intelligence and Robotic Control, vol. 5, no. 2, pp. 33–44. 11. Alexander R. McN. (1984) The Gaits of Bipedal and Quadrupedal Animal. The Int. J. of Robotics Research, vol. 3, no. 2, pp. 49–59.
