Sage Journals: Discover world-class research

Abstract

Inspired by biological systems, we seek to achieve natural dynamics and versatile locomotion for hopping or running robots by installing a series elastic actuator (SEA) in the joints due to its compliant property, passive adaptability and energy storage. However, robots equipped with these actuators have drawbacks in terms of substantial delay and limited bandwidth in their position control, especially when a robot has to choose its foothold while it is running at a demanding speed. To solve these problems, compliance control and adaptive position/torque control are introduced to a hopping- legged robot in this paper. The compliant performance of the robot can be improved through the intrinsic property of an SEA with a torque control algorithm. Combining the kinetics model and stochastic model of a 2-DOF robot, an adaptive position control with Kalman Filtering (KF) is developed to provide rapid convergent state estimation of the load on the robotic end-effector by solving the inverse dynamics. Validating the robustness and effectiveness of the proposed algorithm on our hopping-legged robot Tigger, the experimental results show very good position-tracking and disturbance-rejection, as well as flexible interactions while operating in a complex environment.

Keywords

Legged Robot Compliant Adaptive Series Elastic Actuator (SEA)State Estimation

1. Introduction

In order to study and understand the principles of human and animal locomotion, lots of legged systems have been developed in the last twenty years. Roughly, these robots can be separated into two classes [1]. One class is of stiff-legged systems, which often have a rigid structure and many degrees of under-actuation, as well as use inertia driven movements to propel themselves forward. The rule for designing and controlling robots is ‘the stiffer the better’, which not only improves precision, stability and bandwidth of position-control [2], but also presents higher locomotion speed and lower vibration [3 –6].

The other class of legged systems relates to those equipped with a force or torque sensor, such that a joint or the structure allows the robot to exploit natural locomotion with compliant control. There are two main methods to realize this feature. One is active compliance control of rigid bodies, in which a force sensor is mounted for real-time detection of external contact force on the robot joints or surface. Active compliant control is used to adapt the collision or impact, such as impedance control [7], position and force control [8], and current control [9]. For instance, the quadruped robots BigDog [10] and MIT-Cheetah [11], the humanoid robots COMAN [12] and KONG [13], and the single-legged robot Kenken [14] offer good stability and versatility. However, impedance control is difficult to achieve accurately in either position or force control, because position or force control has to switch mode according to different conditions, while current control needs the precision model of a robot.

Inspired by biology, an SEA is proposed by installing an elastic element between the gear unit and the load [2]. Compared with traditional joints, it has some advantages, such as precise force or torque control, low impedance and elastic energy storage [15,16], which have been widely used in some robots [17 –19, 26]. In term of torque control, some of the literature has been expounded [22 –24]. Besides torque control, position control also plays an important role in improving the performance of SEA-based robots in choosing the foothold or tracking the planned trajectory for versatile motions with visual sensors [27]. Lv [20] described an adaptive method for the position control in an SEA platform, while Zhu [21] designed a BP neural network to compensate for the non-linear characteristics of the spring of an SEA to realize the stable velocity control. But these research studies were only concerned with one joint.

For our research, we extended the principle of high compliance actuators, including non-linear spring and damping characteristics, as well as sophisticated low-level control to enable high fidelity torque control and precise position control. This paper presents a 2-DOF articulated leg robot called Tigger, which has been successfully developed for hopping and running experiments.

This paper is organized as follows. Section 2 introduces the basic structure of an SEA and also describes a torque control method. Section 3 establishes a dynamic model of a single-leg robot and proposes an adaptive position control method with KF. Section 4 demonstrates the leg platform and experimental results. Finally, in Section 5, conclusions and future work are discussed.

2. Compliance Control

The basic principle of an SEA is shown in Figure 1. The main difference between an SEA and a traditional actuator is that an elastic element is embedded between the gearbox and its load in the former, in which the deformation of the elastic element multiplied by its elasticity is equal to the output torque T_l as a feedback to the desired T_d.

Figure 1.

The block diagram of closed-loop feedback torque controller of an SEA [14], where $θ_{g b}, θ_{l}$ is the angle output of gearbox and load, while $T_{d}, T_{l}$ are respectively the desired torque and actual output torque

Consider the following damping term:

k_{s} (θ_{g b} - θ_{l}) + d_{s} ({\dot{θ}}_{g b} - {\dot{θ}}_{l}) = T_{l}

(1)

where d_s is the damping coefficient. In order to improve the precision of the output torque, the velocity loop of the motor is used as the inner loop of the torque controller. When considering the reduction ratio s of the gearbox and the desired motor speed $ϖ_{m d}$ , an equation is obtained after a simple deduction:

\frac{ϖ_{m d}}{n_{g b}} = \frac{T_{l} s}{k_{s} + d_{s} s} + θ_{l} s

(2)

When considering $θ_{l} s$ and $\frac{T_{l} s}{k_{s} + d_{s} s}$ as the feed-forward compensation term of desired motor speed, the former is responsible for tracking the movement of the load, while the latter is used to compensate for the spring deflexion. An SEA torque control is then implemented with a PID controller, which is the control block diagram shown in Figure 2.

Figure 2.

Diagram of the torque control framework

Also in Figure 2, the controller structure introduces the cascaded structure with current control and velocity control on a motor controller and a torque controller, as well as a position controller, which will be discussed later. Assuming the controller is ideal, the expected torque T_d is treated instead of the real output torque $T_{l}$ as the feedforward term. In addition, for the purpose of reducing the high-frequency noise, low-pass filters are added on some differential items. Torque control compensates the errors caused automatically by backlash and friction based on the velocity loop of the motor. The inverse dynamics compensation term and feedforward term obtained by analysing an SEA model decrease the proportion of PID output in the main loop. Since the PID output contains only the error correction term caused by the unmodelled dynamics and random noise, the approach can increase the control gain, speed up the response and improve the ability resistance to disturbance. Moreover, the compliance of the actuator can be changed by adjusting the output torque threshold of an SEA, so that the contact or impact between an SEA-based actuator and environment can be controlled in the safety range. Furthermore, the elasticity coefficient k_s defines the upper bound of compliance with the mechanical structure.

3. Adaptive Position Control

The passive compliance of an SEA reduces risk when the robot interacts with an unknown environment. However, the introduction of the elastic element easily leads to the oscillation of the load, which increases the difficulty of position control. Moreover, the stiffness of the elastic element itself is unable to adapt to the condition while changing the load. In this paper, an effective solution based on the stochastic model of adaptive position control is described in order to quickly identify the damping and load of the SEA-based robot once they have been changed. It can effectively eliminate the oscillation, achieve a fast accurate position control and ensure operating safety in an unknown environment. Besides, by adjusting feedback gain of position control based on the torque loop, it can easily change the practical manifestation of output impedance when it works in different forms of movement or suffers from disturbance due to different frequency and size.

3.1. Adaptive control

The single-leg model mainly consists of three linkages: body, thigh and shank. The SEA is mounted in both the hip and knee joints where torque and position control can be introduced, which is the main focus of this paper. As this paper does not refer to the balance control of a single- leg robot, it can be assumed that the body is fixed with only the leg that is able to move.

The model of the single-leg robot can be seen in Figure 3. $θ_{1}, θ_{2}$ are respectively the hip and knee angles, while $m_{1}, m_{2}$ relate to the mass of link 1 and link 2, which represents the thigh and shank of the leg, while $I_{1}, I_{2}$ is the inertia. The displacements of the centre relative to the hip and knee joints are $d_{1}, d_{2}$ , while $δ_{1}, δ_{2}$ concerns the angle between the CoM of the thigh and knee to their respective joints. When considering damping, disturbance and unmodelled dynamics, the dynamics equation of the robot is obtained by Lagrange equations:

H (q) \ddot{q} + C (q, \dot{q}) \dot{q} + g (q) + f (q, \dot{q}, t) + D_{s} (t) = T_{u}

(3)

where $H (q)$ is the inertia momentum of the leg, $C (q, \dot{q})$ is the Coriolis matrix and $g (q)$ is the gravity matrix:

\begin{array}{l} H (q) = [\begin{matrix} a_{1} + a_{2} \cos θ_{2} & a_{3} + a_{2} \cos θ_{2} \\ a_{2} \cos θ_{2} & a_{3} \end{matrix}] \\ C (q, \dot{q}) = [\begin{matrix} a_{2} \sin θ_{2} \cdot {\dot{q}}_{1} & - a_{2} \sin θ_{2} {\dot{q}}_{2} \\ a_{2} \sin θ_{2} \cdot {\dot{q}}_{1} & 0 \end{matrix}] \\ g (q) = [\begin{matrix} a_{4} \sin q_{1} + a_{5} \sin q_{2} \\ a_{5} \sin q_{2} \end{matrix}] \end{array}

$f (q, \dot{q}, t)$ is the torque caused by friction and other unmodelled dynamics, and $D_{s} (t)$ is the joint torque produced by disturbance.

When the upper bound of torque error caused by kinetic modelling and disturbance is set at ψ, then we have:

| f (q, \dot{q}, t) | + | D_{s} (t) | \leq ψ

(4)

Calculated by Lagrange dynamic equations, we can obtain the unknown parameters to be estimated as a vector:

a = {[\begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \end{matrix}]}^{T}

(5)

where

\begin{array}{l} a_{1} = I_{0} + I_{2} - I_{3} = m_{1} d_{1}^{2} + I_{1} + m_{2} l_{1}^{2} \\ a_{2} = I_{1} = m_{2} l_{1} d_{2} \\ a_{3} = I_{3} = m_{2} d_{2}^{2} + I_{2} \\ a_{4} = G_{1} = (m_{1} d_{1} + m_{2} l_{1}) g \\ a_{5} = G_{2} = m_{2} d_{2} g \end{array}

\begin{array}{l} I_{0} = m_{1} d_{1}^{2} + I_{1} \\ I_{1} = m_{2} l_{1} d_{2} \\ I_{2} = m_{2} l_{1}^{2} + m_{2} d_{2}^{2} + I_{2} \\ I_{3} = m_{1} d_{2}^{2} + I_{2} \\ G_{1} = (m_{1} d_{1} + m_{2} l_{1}) g \\ G_{2} = m_{2} d_{2} g \end{array}

The augmented error can now be defined as:

s = \dot{\tilde{q}} + Λ \tilde{q} = \dot{q} - {\dot{q}}_{r}

(6)

Here, $\tilde{q} = q - q_{d}$ , q_d refer to the desired joint angles, while $Λ = [\begin{matrix} \begin{matrix} λ_{1} & 0; \end{matrix} & \begin{matrix} 0 & λ_{2} \end{matrix} \end{matrix}]$ is a $R^{2 \times 2}$ positive diagonal matrix, which needs to be set in the controller in the same way as the PID parameters. ${\dot{q}}_{r} = {\dot{q}}_{d} - Λ \tilde{q}$ is the virtual desired speed.

Through feedback decoupling linearization, the dynamics model can be expressed as below:

\begin{array}{l} H {\ddot{θ}}_{r} + C {\dot{θ}}_{r} + g θ + \\ f (θ, \dot{θ}, t) + D (t) = Y (θ, \dot{θ}, {\dot{θ}}_{r}, {\ddot{θ}}_{r}) a \end{array}

(7)

If $\hat{a}$ is the estimation of the unknown parameters a, then the following control law is derived:

T_{u} = ϒ (θ, \dot{θ}, {\dot{θ}}_{r}, {\ddot{θ}}_{r}) \hat{a} - k_{D} s

(8)

Control law T_u consists of the dynamic feedforward term $Y (θ, \dot{θ}, {\dot{θ}}_{r}, {\ddot{θ}}_{r}) \hat{a}$ and its error compensation $k_{D} s$ . If the error compensation $| k_{D} s | < ψ$ , the former dynamic model is sufficient to describe the robot joint torque, while the parameter estimation is close to convergence. Therefore, further estimation is not needed in order to update these parameters; otherwise, the parameters need to be estimated further.

As such, apply the adaptive parameter estimation update laws:

\dot{\hat{a}} = - Γ ϒ^{T} s_{ψ}

(9)

where $Γ = d i a g {γ_{1}, γ_{2}, γ_{3}}$ is the update rate of the estimated parameter:

s_{ψ} = s - ψ k_{D}^{- 1} s a t (k_{D} ψ^{- 1} s), s a t (x) = {\begin{matrix} x & | x | < 1 \\ 1 & x \geq 1 \\ - 1 & x \leq - 1 \end{matrix}

(10)

Parameter update laws $\dot{\hat{a}}$ with a dead zone can effectively decrease instability caused by small perturbation, as well as increase the parameters' convergence rate of the system.

Consider the following Lyapunov function:

V (t) = \frac{1}{2} [s_{ψ}^{T} H (q) s_{ψ} + {\tilde{a}}^{T} Γ^{- 1} \tilde{a}]

(11)

then

\begin{array}{l} \dot{V} (t) = s_{ψ}^{T} [H \ddot{q} - H {\ddot{q}}_{r}] + \frac{1}{2} s_{ψ}^{T} \dot{H} s_{ψ} + {\dot{\hat{a}}}^{T} Γ^{- 1} \tilde{a} \\ = s_{ψ}^{T} [T_{u} - H {\ddot{q}}_{r} - C {\dot{q}}_{r} - g] + (\frac{1}{2} s_{ψ}^{T} \dot{H} s_{ψ} - s_{ψ}^{T} C s_{ψ}) \\ \leq s_{ψ}^{T} [T_{u} - H {\ddot{q}}_{r} - C {\dot{q}}_{r} - g] + {\dot{\hat{a}}}^{T} Γ^{- 1} \tilde{a} \end{array}

(12)

substituting (8) into (12):

\begin{array}{l} \dot{V} (t) \leq - s_{ψ}^{T} k_{D} s \leq 0 \\ \ddot{V} (t) \leq - 2 s_{ψ}^{T} k_{D} {\dot{s}}_{ψ} \end{array}

(13)

According to Barbalat theory [25], the global stability and tracking error convergence of the adaptive control system can be proven by Equation (13). The control system considers the load as a part of the robot, thereby estimating the load mass to calculate the torque of the joint. Here, the estimated parameters $\hat{a}$ correspond with the physical parameter $m_{i}, d_{i}, I_{i}, i \in [\begin{matrix} 0 & 1 \end{matrix}]$ , which simplifies the design of the control system. Based on the settings of the threshold of output torque, position control can ensure safety and compliance as well as provide tracking in complex environments.

3.2. Introduction of KF state estimation

An adaptive compliant position control of a robotic leg has been implemented. However, the update rate of adaptive estimation parameter Γ is given empirically; therefore, if it is set too large, the stability margin of the system will decrease while, if it is too small, it will be too slow to update the parameters and the capability of real-time performance will decline. Meanwhile, Γ cannot be adjusted automatically because the load changes require a long adaptation time.

KF predicts errors and calibrates estimates by a non-linear system and observation. It can overcome many effects of external uncertain factors. With regard to the changed loads of a robotic leg as one of the uncertainties, the KF state estimator is added to an adaptive SEA position control. Combined with observed value and state prediction, optimal value will be estimated from the current state. It makes the parameter update rate vary according to the load change state and improves the adaptation of the parameter estimation rate.

Take the adaptive estimated parameters $\hat{a}$ as the state of KF $\hat{a} (t) = {[\begin{matrix} {\hat{a}}_{1} (t) & {\hat{a}}_{2} (t) & {\hat{a}}_{3} (t) & {\hat{a}}_{4} (t) & {\hat{a}}_{5} (t) \end{matrix}]}^{T}$ . Due to a lack of sensors to observe the estimated parameters $\hat{a}$ , in order to build the motion model with insufficient information, the equation can be described by:

\hat{a} (k + 1) = \hat{a} (k) + v_{t}, v_{t} \sim N (0, R_{t})

(14)

where R_t is the uncertainty of motion. Given the coupling of KF state variables, it cannot be regarded as an independent state variable to estimate directly, otherwise it would mismatch the KF state estimation with an adaptive parameter update, even in the opposite direction. The changes in the load will be reflected on several dynamics and independent physical parameters,

such as $m_{i}, d_{i}, C$ , which determine the variance and covariance of motion equation; R_t reflects the coupling effects between these variables.

By regarding the controller output torque as the actual observed data, and the output of the adaptive feedforward term as observed prediction according to the current state, the observation equation is:

u = ϒ \hat{a} + w_{t}, w_{t} \sim N (0, Q_{t})

(15)

where Q_t is the uncertainty of observation. The error between observation and prediction is $- k_{D} s$ . There are some reasons for this: (1) although there are some differences between the adaptive feedforward model $Y \hat{a}$ and the real dynamics model, the actual output of the controller u, which is corrected by feedback, makes $Y \hat{a}$ as close to the real dynamics leg model (including unmodelled dynamics) as possible; (2) although the controller output T_u differs greatly compared to the required torque in the initial stage, the adaptive feedforward model approaches the real dynamics model of the leg by correcting the system status with KF observation. Therefore, the feedforward model would guarantee convergence with the real dynamics model of the robot by treating the controller output as the observed value of KF observation equations.

Due to the contradictions in anti-disturbance and convergence rates between the motion model and observations model with constant variance, an adaptive method for determining $R_{t}, Q_{t}$ is proposed. The decision of the status observer is more inclined to the observation model or motion model depending on the estimation of $R_{t}, Q_{t}$ .

Uncertainty about the observation model is determined by measuring the noise of the control function. Due to the frequency band of the controller output being far away from the noise, noise generated from the high-frequency component of the controller through a rapid Fourier transformation determines Q_t. Uncertainty of the motion model is determined by the error between the actual output of the controller and the adaptive feedforward item. The uncertainty of the motion model R_t can be represented by decoupling the output calibration term from the state estimation error $- k_{D} s$ .

Using the KF equations, the optimal state estimation can be derived as:

\hat{a} (k | k) = \hat{a} (k | k - 1) + K_{t} (u - ϒ \hat{a} (k | k - 1))

(16)

Here, $K_{t} = {\bar{Σ}}_{t} Y^{T} {(Y {\bar{Σ}}_{t} Y^{T} + Q_{t})}^{- 1}$ is the gain matrix of the filter, while ${\bar{Σ}}_{t}$ is the covariance matrix. After transforming the KF state estimation into an adaptive parameter $\hat{a}$ , the increments are:

Δ \hat{a} = \frac{{\bar{Σ}}_{t} k_{D}}{ϒ {\bar{Σ}}_{t} ϒ^{T} + Q_{t}} ϒ^{T} s

(17)

Compared with the update rate of an adaptive controller after dealing with a dead zone, the update rate of an estimated parameter can be improved by constant to real-time adjustment based on the confidence of the motion model and observation model; the overall control system block diagram is shown in Figure 4.

Figure 3.

A simplified model of a 2-DOF articulated leg

Figure 4.

The diagram of adaptive control within a KF framework

If the load of the robot remains unchanged, the feedforward convergence value of parameter estimation is sufficient to provide the input of a controller and will not appear as a significant error. Meanwhile, the uncertainty of the kinematic equation R_t will reduce, such that the system becomes more dependent on the motion model and, thus, the update rate slows down. Moreover, the lower update rate and KF curb the effects of transient disturbances and improve the robustness of the system. If the load experiences sudden change, the parameter estimated value is impossible to meet the real demands of adaptive feedforward, and in turn the deviation of position appears. In this case, uncertainty Q_t will reduce while R_t increases, with the control system becoming more dependent on the observation model. By increasing the update rate greatly, the adaptive feedforward control term will get closer to the real dynamics of an SEA-based robot in a shorter time, in order to satisfy the purpose of control.

4. Experimental Results

4.1. Experiment set-up

In respect of the mechanical design, our robot is composed of three linkages: thigh, shank and body. Two articulated SEA joints are equipped with a knee and a hip joint, while a passive joint is mounted on the body to constrain the motion in the sagittal plane. An SEA joint, due to the elastic element embedded into it, significant reduces the response bandwidth compared with a traditional rigid joint. In fact, humans have a similar bandwidth in their joints with a slightly larger range of movement, at 10Hz, for walking, running and crawling. In order to maximize the benefit of fast response and low energy loss, the CoG of the whole robot is designed to be close to the axis of the hip joint, while a parallel four-bar linkage mechanism is employed for the transmission motion of knee, which requires the knee motor to be placed close to the body. This design also brings with it an additional benefit of convenience in being able to adjust the knee joint's stiffness by simply replacing the outer springs; however, we do not use this function in our study. A carbon fibre tube is installed on the body with an extra load at both ends in order to increase the inertia ratio between body and leg. In this paper, an extra load was installed on the foot. The structure of the single-leg robot is shown in Figure 3, while a photograph of the actual leg robot, Tigger, is shown in Figure 5.

Figure 5.

Platform of the SEA-based single-leg robot Tigger

In respect of the electrical design, two identical 200W 4Pole Maxon motors combined with 120:1 CSG harmonic drivers are used in both the knee and hip joints. Each joint is equipped with two 18-bit absolute encoders to measure the deformation of the spring. Distributed circuit boards are inscribed on the ARM processors for controlling each motor of the joint independently, in order to reduce the burden of the CAN bandwidth and receive the planning data from the PC controller through a CAN bus. An MTi-100 IMU is installed at the centre of the body to measure its posture by USB, while a switch sensor is installed at the end of the foot for judging whether the foot touches the ground or not, which will be used for the hopping experiments.

Table 1.

Physical parameters of the single-leg robot

Symbol	Description	Value	Unit
P	Rated output power of each motor	200	Watts
n_g	Reduction ratio of each joint	120:1	--
k_h	Spring elasticity in hip	212.1	Nm/rad
k_k	Spring elasticity in knee	172.3	Nm/rad
ϖ	Large force bandwidth	10	Hz
l₁	Length of robotic link 1	0.32	M
l₂	Length of robotic link 2	0.32	M
m₁	Mass of thigh link from CAD	3.23	Kg
m₂	Mass of shank link from CAD	0.53	Kg
Δm	Mass of extra load on the foot	1.0	Kg
I₁	Movement of inertia of thigh from CAD	0.04	kgm²
I₂	Movement of inertia of shank from CAD	0.015	kgm²
t_p	Period cycle of position control	8	Ms

4.2. Experimental results

Zero-torque control mode plays an important role in legged-robot locomotion as a special case of torque control. This is an important characteristic of an SEA joint because it makes it possible for the joint to work in either a compliant active or passive pattern mode. Especially when it works in the passive mode, the joint can work as a natural pendulum with a little mechanical damping due to its gravity or external thrust. This mode may be used for passive locomotion or energy storage in future. The curves are shown in Figure 6.

Figure 6.

Zero-torque control in the hip and knee joints

According to Figure 6, it can be seen that the actual torque oscillated around zero. Due to the limitation of the control period cycle and motor response time, the output cannot follow the desired zero-torque absolutely. But we can find that the knee joint has a better tracking performance than the hip one, because the knee has a lower spring elasticity, which not only increases the flexibity of the joint itself, but reduces the response time requirements of the motor.

In the position control experiment, adaptive position control with and without KF has been compared, as shown in Figures 7 and 8. It can be seen that control with KF needs less time to track the desired position than control without KF. Due to the inaccuracy of the adaptive parameter estimation at the beginning, some shocks occur in a short time due to the adaptive position control. Once the estimated parameter converges, good tracking effects appear.

Figure 7.

Hip position control without and with KF

Figure 8.

Knee position control without and with KF

The parameter convergence rate of control method with and without KF is presented in Figure 9. As we can see, at the same initial parameters, after adding the sudden load, the former parameter convergence takes about four seconds, while the latter takes only two seconds and the convergence rate increases by about 50%.

Figure 9.

Comparison of convergence rates without and with KF

Figure 9 shows the tracking performance of adaptive control with KF while the load changed. This method maintains high precision in tracking performance.

Compliance and disturbance rejection were tested by artificially induced disturbance while SEA joints were in position-tracking mode. As seen in Figure 10, the adaptive compliant control system has a good performance in terms of disturbance rejection and maintains a high level of stability. This shows that the robotic leg continues to deal with the disturbance rather than hurting like a human or damaging the motor with a path constraint. The locomotion of the single-leg robot immediately recovers to an accurate position-tracking after release from the disturbance without switching to any control mode. The compliant SEA-based robot, therefore, has the capabilities for precise position-tracking and security interaction in complex and unknown environments.

Figure 10.

Disturbance resistance and compliance test

5. Conclusion

In this paper, an adaptive compliant position control method with KF is proposed. The main advantages of this method are as follows: (1) based on the dynamic model of the single leg, the adaptive control law implements compliance position control and guarantees operational safety; (2) introducing a stochastic model can effectively counterbalance the effect of disturbance and improve the robustness of the robot system; and (3) treating KF as part of the adaptive learning rate improves the convergence rate of an adaptive parameter in response to rapid load change. The experimental results show us that the methods proposed in this paper can provide a good position-tracking, disturbance-rejection and flexible interactions while operating in complex environments. It serves as a strong foundation for our future work in pursuit of more automated and natural locomotion of humanoid robots, quadruped robots and manipulators.

Footnotes

6. Acknowledgements

This research is based upon work supported in part by the National Nature Science Foundation of China (514054530, 61473258, 61273340) and the Public Welfare Technology Application Research Plan of Zhejiang (2016C33G2010137).

References

Hutter

Remy

C. D.

Hoepflinger

M. A.

. (2011) ScarlETH: Design and control of a planar running robot. IEEE/RSJ International Conference on Intelligent Robots and Systems: 562–567.

Pratt

G. A.

Williamson

M. M.

Dillworth

. (1995) Stiffness isn't everything. Preprints of the Fourth International Symposium on Experimental Robotics: 253–262.

Sakagami

Watanabe

Aoyama

. (2002) The intelligent ASIMO: System overview and integration. Proceedings of the 2002 IEEE/RSJ International Conference on Intelligent and Systems: 2478–2483.

Kaneko

Kanehiro

Morisawa

. (2011) Humanoid Robot HRP-4: Humanoid robotics platform with lightweight and slim body. IEEE/RSJ International Conference on Intelligent Robots and Systems: 4400–4407.

Buschmann

Ewald

Ulbrich

. (2012) Event-based walking control form neurobiology to biped robots. IEEE/RSJ International conference on Intelligent Robots and Systems: 1793–1800.

Zucker

Joo

Grey

M. X.

. (2015) A general purpose system for teleoperation of the DRC-HUBO humanoid robot. Journal of Field Robotics: 32(3), 336–351.

Hogan

. (1984) Impedance control: An approach to manipulation. In American Control Conference: 304–313.

Raibert

M. H.

Craig

J. J.

. (1981) Hybrid position/force control of manipulators. Journal of Dynamic Systems, Measurement, and Control: 103(2), 126–133.

Zhang

Ahman

Liu

. (2015) Torque estimation for robotic joint with harmonic drive transmission based on position measurements. IEEE Transaction on Robotics: 31(2), 322–330.

10.

Buehler

Playter

Raibert

H. H.

. (2005) Robots step outside. Int. Symp. Adaptive Motion of Animals and Machines.

11.

Seok

Wang

Otten

. (2012) Actuator design for high force proprioceptive control in fast legged locomotion. IEEE/RSJ International Conference on Intelligent Robots and Systems: 1970–1975.

12.

Zhou

Tsagarakis

. (2015) Compliance control for stabilizing the humanoid on the changing slope based on terrain inclination estimation. Autonomous Robot, springer, doi: 10.1007/s10514-015-9504-6.

13.

Xiong

Sun

Zhu

. (2012) Impedance control and its effects on a humanoid robot playing table tennis. International Journal of Advanced Robotic Systems, Vol. 9, 178:2012.

14.

Hyon

S. H.

Mita

. (2002) Development of a biologically inspired hopping robot – Kenken. IEEE/RSJ International Conference on Robotics and Automation: 3984–3991.

15.

Robinson

D. W.

. (2000) Design and analysis of series elasticity in closed-loop actuator force control. Massachusetts Institute of Technology.

16.

Paluska

Herr

. (2006) Series elasticity and actuator power output. IEEE/RSJ International Conference on Robotics and Automation: 1830–1833.

17.

Tdsgarakis

Galdwell

. (2013) Walking pattern generation for a humanoid robot with compliant joints. Autonomous Robots, 35(1), 1–15.

18.

Pratt

Krupp

. (2011) Design of a biped walking robot. in Unmanned Systems Technology X, 69612F_1-69612F-13.

19.

Hutter

Remy

C. D.

Hoepflinger

M. A.

. (2011) ScarlETH: Design and Control of a planar running robot. IEEE/RSJ International Conference on Intelligent Robots and Systems: 562–567.

20.

Zhu

Xiong

. (2014) An adaptive compliance position control based on EKF for series elastic actuation. Proceeding of the 11th World Congress on Intelligent Control Automation: 3100–3106.

21.

Zhu

Xiong

. (2015) Novel series elastic actuator design and velocity control. Electric Machines and Control, 19(6), 83–88.

22.

Pratt

G. A.

Willisson

Bolton

. (2004) Late motor processing in low-impedance robots: Impedance control of series-elastic actuators. In American Control Conference: 3245–3251.

23.

Kong

Bae

Tomizuka

. (2012) A compact rotary series elastic actuator for human assistive systems. IEEE/ASME Transaction on Mechatronics, 17(2), 288–297.

24.

Kong

Bae

Tomizuka

. (2009) Control of rotary series elastic actuator for ideal force-mode actuation in human–robot interaction applications. IEEE/ASME Transaction on Mechatronics, 14(1), 105–118.

25.

Slotine

J. E.

. (2004) Applied Nonlinear Control. China Machine Press.

26.

Tsagarakis

N. G.

Caldwell

D. G.

. (2012) Walking trajectory generation for humanoid robots with compliant joints: experimentation with COMAN humanoid. IEEE/RSJ International Conference on Robotics and Automation: 836–841.

27.

Liu

Xiong

. (2014) Robust and accurate multiple-camera pose estimation toward robotic applications. International Journal of Advanced Robotic Systems, 11:153, doi: 10.5772/58868.

Adaptive Torque and Position Control for a Legged Robot Based on a Series Elastic Actuator

Abstract

Keywords

1. Introduction

2. Compliance Control

3. Adaptive Position Control

3.1. Adaptive control

3.2. Introduction of KF state estimation

4. Experimental Results

4.1. Experiment set-up

4.2. Experimental results

5. Conclusion

Footnotes

6. Acknowledgements

References