An optimal jerk-stiffness controller for gait pattern generation in rough terrain
- Amira Aloulou^{1}Email author and
- Olfa Boubaker^{1}
Received: 26 November 2015
Accepted: 14 June 2016
Published: 11 July 2016
Abstract
In this paper, an optimal jerk stiffness controller is proposed to produce stable gait pattern generation for bipedal robots in rough terrain. The optimal jerk controller is different from the point-to-point and via-Point conventional approaches as trajectories are planned in the Cartesian space system whereas control laws are expressed in the joint space. Its major contribution resides in the generation of stable semi elliptic Cartesian trajectories during the swing phase that do combine benefits of trigonometric and polynomial functions. The stiffness controller is designed without gravity compensation and ensures for the robotic system elastic and stable contact forces with the ground during the impact and the double support phases. Not only, the control strategy proposed needs very few sensors to be implemented but also it ensures robustness to sensory noise and safety with rough terrain. Simulation performed on a 12 DOF bipedal robot shows the performances of the control laws combined to produce a 3D stable walking cycles without shaking in uneven terrain.
Keywords
Background
Prospective gait pattern generation is one of the main challenges of research dedicated to bipedal and humanoid robots. A typical walking cycle includes three main stages: the single support phase (SSP), the impact phase (IP) and the double support phase (DSP). The SSP occurs when one limb is pivoted to the ground while the other is swinging from the rear to the front. The IP occurs when the toe of the forward foot starts touching the ground. The impact between the toe of the swing leg and the ground takes place during an infinitesimal length of time. Finally, the DSP occurs when both limbs remain in contact with the ground. During the SSP, the robotic system is described by a free dynamic model while IP and DSP phases represent the constrained dynamic model.
In the last decades, many control techniques have been investigated to produce human-like walking gaits mainly based on the inverted pendulum principle [1]. During the SSP, several optimal control laws have been also proposed. Criteria to be optimized are often the energy consumption [2], the falling measure [3], the ZMP [4] and the jerk [5]. Focusing on the jerk criterion, the correspondent optimal control law has many benefits. Its main involvement resides in the generation of smooth motion trajectories in order to avoid sudden movements [6]. The optimal jerk based control techniques have affected many industrial areas such as machine tools, manufacturing, and robotics [7–11]. However, for the selection of mathematical function to describe desired trajectories to be tracked, there are divided opinions between researches selecting polynomial functions [12] and, others using trigonometric functions [13, 14]. For the first case, it is proved that polynomial trajectory references are easily followed by the actuators involved. However, such approaches have certain disadvantages like the long execution time which seriously jeopardizes the possibility of real time implementation. For the second case, it is proved that the involved joints in the movement are less oscillatory when trigonometric functions are considered. Another issue raised has also divided researchers in opinion: the space on which reference trajectories must be planed (Cartesian or joint space). For example, it is showed in [15] that if the minimization problem and its solution are formulated in the joint space, only physical limitations of the joint actuators will be included in the constraint statements. However, in a realistic environment, obstacles exist and cause changes in the trajectory direction. Actually, generating reference trajectories may be done whether in the joint or Cartesian space. The space’s choice should only be determined according to the constraints and the shape of the desired trajectory.
On the other hand, to produce stable and safe contact with the ground during the IP and DSP, a number of control techniques can be used to solve the force/position control problem [16]. The active stiffness approach, originally proposed by Salisbury in [17], can be adopted. Its goal is to establish a dynamic relation between the end-effector position and the contact force. Such control approach has the advantage to provide elastic contact with the constrained environment while requiring very little sensors. Unfortunately, it generally suffers of lack of precision and robustness. Recently, several research papers have proposed some improvements to overcome such problems [18–20].
In this paper, the major contribution lies in the proposal of two control laws combined to produce 3D safe walking cycles: An optimal jerk controller during the SSP and an active stiffness controller law without gravity compensation during the constrained phases. The resulting control approach guarantees a stable and safe gait pattern generation without vibration and shaking even in presence of sensory noise and rough terrain.
This paper is organised as follows: In ‘‘The robotic model” section, the robotic model during the SSP, the IP and the DSP is described. ‘‘The 3D desired trajectory of the swing foot” section presents the 3D desired trajectory of the swing foot during the SSP. Jerk optimal control and Stiffness control laws are designed in ‘‘Jerk optimal control” section and ‘‘Active stiffness controller” section, respectively. Finally, simulation results performed on a 12 DOF bipedal robot are given in ‘‘Simulation results” section.
The robotic model
The 3D desired trajectory of the swing foot
Jerk optimal control
Minimum jerk principle
To compute the parameters a _{0} … a _{5}, two main methods are used in the literature: The Point-to-point method [22] and the Via-point one [23]. The first methodology only requires the expression of the function to be minimized and the values of position, velocity and acceleration of the initial and final time of the movement. The corresponding control algorithm only needs to run once. For each joint, the following relation is used [22]:
The control law
As the dynamic modeling of the bipedal robot is known, \(K_{p,1} {\text{and }}K_{v,1}\) are computed offline to satisfy global stability conditions.
The jerk optimal algorithm
- i.
For desired initial and final Cartesian positions of the toe of the swing foot, compute the initial joint position vector ϕ _{ in } and the final joint position vector ϕ _{ f } using the inverse kinematic model (5).
- ii.
Generate the polynomial trajectories \(\phi_{a}\), \(\dot{\phi }_{a}\) and \(\ddot{\phi }_{a}\) described by (18–20) using the Point-to-Point method according to the relation (11).
- iii.
- iv.
Generate the desired joint trajectories \(\phi_{d}\), \(\dot{\phi }_{d}\) and \(\ddot{\phi }_{d}\) using (4–6).
- v.
Compute the jerk optimal control law U(t) using (26).
- vi.
Implement the control law U(t) for the free robotic system described by the dynamical model (1).
- vii.
Generate the Cartesian trajectories Xt) and \(\dot{X}\)(t) by applying the direct kinematic model (2) and the differential kinematic model (3). ϕ(t) and \(\dot{\phi }\)(t) are supposed to be measured via online sensors.
The differences between the proposed approach and the conventional ones can be summarised as follows: first, the reference trajectory of the swing foot is planned in the Cartesian space with constraints on positions, velocities and accelerations at every time iteration whereas for the point-to-point method constraints are to be found only at the boundary conditions and for the via-point method these constraints must also include intermediary positions, their velocities and their accelerations. Moreover, the proposed approach uses a trigonometric expression of desired trajectories that depends on a fifth order polynomial instead of just having recourse to a fifth order polynomial as done for the conventional approaches.
The proposed method of optimal jerk control is designed in order to reduce significantly the time of implementation. As trajectories are planned in the Cartesian space system whereas control laws are expressed in the joint space, it does combine benefits of trigonometric and polynomial functions. Indeed, trigonometric functions require fewer resources for real time implementation whereas polynomial functions give smoother dynamics and fewer vibrations.
Active stiffness controller
The control law (31) is then designed such that the common gravity compensation term found in many research works in the literature is eliminated thanks to the contact force virtual model proposed in (28). This further gives more robustness to the control law. The control law (31) has also the advantage to reduce the number of sensors. To be implemented, it is clear that only position and velocity sensors are needed.
Finally, to achieve a walking cycle and produce alternate footsteps, the minimum jerk controller (26) and the active stiffness controller (31) are combined to switch successively between the SSP described by the robotic model (1), and the IP and DSP described by the constrained robotic model (8).
Simulation results
Physical parameters of the robot
Link | k _{ i } (m) | l _{ i } (m) | m _{ i } (Kg) | Inertia about center of mass (Kg m^{−2}) | ||
---|---|---|---|---|---|---|
i _{ ix } | i _{ ix } | i _{ ix } | ||||
Right foot | 0.034 | 0.034 | 1.015 | 0.001 | 0.001 | 0.001 |
Right leg | 0.184 | 0.241 | 3.255 | 0.051 | 0.051 | 0.051 |
Right thigh | 0.184 | 0.240 | 7.000 | 0.113 | 0.113 | 0.113 |
Pelvis | 0.021 | 0.178 | 9.940 | 0.112 | 0.112 | 0.112 |
Left thigh | 0.240 | 0.184 | 7.000 | 0.113 | 0.113 | 0.113 |
Left leg | 0.241 | 0.184 | 3.255 | 0.051 | 0.051 | 0.051 |
Left foot | 0.034 | 0.034 | 1.015 | 0.001 | 0.001 | 0.001 |
Parameters of the cartesian desired trajectory
a (m) | b (m) | c (m) | d (m) | (u, v) (m) |
---|---|---|---|---|
0.15 | 0.53 | 0.1 | 16.5 | (0.15, 0) |
Simulation results emphasize the efficiency of the control laws as perturbations and rough terrain have no effect on the bipedal robot trajectory. Compared to previous work found in [24], not only fewer sensors are needed in the implementation of the control laws but the semi-elliptical trajectory duration has been also improved to 0.5 s. It corresponds to an enhanced velocity of the bipedal robot of 0.75 m s^{−1}. Also, an elastic and robust contact with the ground is ensured. This was not guaranteed with the impedance control law.
Conclusion
To produce a path similar to the one generated by a human foot when performing a walking cycle by a bipedal robot, we have proposed, in this paper, a specific walking control strategy using an optimal jerk and an active stiffness controllers based only on position and velocity sensors. Simulation results performed on a 12 DOF bipedal robot emphasized the efficiency of the control strategy even in presence of sensory noise and rough terrain and prove the superiority of the new algorithm regarding the step duration and the bipedal velocity compared to previous works.
Declarations
Authors’ contributions
AA proposed the jerk optimal controller and carried out simulation results. OB designed the active stiffness controller. Both authors write the final manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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