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Analysis of Stability and Horizontal Motion of a Single Leg Hopping Robot

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Abstract

The structure of the hopping robot is such that the robot, relying on a single leg, must make consecutive jumps and be able to maintain its balance with hand tools. For this purpose, changing the angle of the robot's hands on the upper body is responsible for maintaining the robot's balance. If the upper body function is optimal, the leg mechanism can jump from the momentum caused by the robot mass. In this research, a semi-active mechanism is used for jumping, and the active actuator only performs the task of compensating for wasted energy. The new content of this research is based on the use of a simple four-link mechanism to compress the passive element rather than the ball-screw mechanism or any complete active mechanisms for jumping actions, and two moving arms to get balanced and do horizontal motion like acrobats by leading forward and changing orientation based on a unicycle model. Robot control method and how to implement PD controllers to move on a plane in a 3D space are presented; also, this paper presents the total modeling and motion control based on jump movement, balance, and robot orientation. The control concept of jumping is obtained as an open-loop multi-phases controller with no feedback from spatial position and orientation, just the internal joints of the robot, as well as the balancing control that used feedbacks of position and velocity states for observing different phases. Although, the rotating motion is a feedforward control. Finally, by implementing the control method and the robot model in a virtual environment (Simscape/Matlab), the robot's pursuit of some paths simulated. So the control motions act independently to guide the robot at a maximum speed of 3.1 cm per second. The simulations included the following online and offline trajectories, and the robot's performance is validated for movements in the range of 7–16 m per 260–600 s, as the robot is able to jump as high as 28 cm at the duration of 0.6 s in each stage hop.

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Data availability

Owing to the fact that the data supporting the findings of this article is going to be utilized for future studies, participants of this research do not agree for their data to be shared publicly.

Abbreviations

CM:

Center of mass (assumed in middle of body)

\(x\) :

Displacement of CM on the direction

\(\dot{x}\) :

Linear velocity of CM

\(z\) :

Height of CM

\(\dot{z}\) :

Linear jumping velocity of CM

ɣ:

Orientation of robot on plane

ɣ̇ :

Rotational velocity of robot on plane

\(\alpha\) :

Angle between link and vertical line

\(\dot{\alpha }\) :

Rate of angle between link and vertical line

\(d\) :

Elongation of spring

\({d}_{1}\) :

Height of toe

\({L}_{0}\) :

Free length of spring

\({L}_{1}\) :

Half of length of arm1

\({L}_{2}\) :

Half of length of arm2

\({L}_{3}\) :

Length of link (four-link mechanism)

ɸ:

Constant angle between arm1 and the direction of the robot body horizon

\(\mathrm\theta\)  :

Angle between arm1 and leg direction

\(\dot\theta\)  :

Rate of Angle between arm1 and leg direction

Ψ :

Angle between CM to leg direction and vertical in x–z plane

ƞ :

Angle between CM to leg direction and vertical in y–z plane

\(\beta\) :

Angle between hand 1and hand 2

\({k}_{0}\) :

Stiffness of linear spring

\({k}_{1}\) :

Rotational stiffness of spring of joint1

\({k}_{2}\) :

Rotational stiffness of spring of joint3

\({c}_{0}\) :

Damping coefficient of linear spring

\({c}_{1}\) :

Rotational damping coefficient of joint1

\({c}_{2}\) :

Rotational damping coefficient of joint3

\({\alpha }_{0}\) :

Free angle of spring of joint1: 90 degree

\({\beta }_{0}\) :

Free angle of spring of joint3: 90 degree

M:

Mass of frame body

\({m}_{0}\) :

Mass of link (each link of four-link mechanism)

\({m}_{1}\) :

Mass of arm1

\({m}_{2}\) :

Mass of arm2

\({m}_{3}\) :

Mass of toe

\({I}_{1}\) :

Moment of inertia of arm1 with respect to the center of mass in x–y plane

\({I}_{2}\) :

Moment of inertia of arm2 with respect to the center of mass in x–y plane

\({I}_{3}\) :

Moment of inertia of links with respect to the center of mass in x–z plane

\({I}_{4}\) :

Moment of inertia of arm1 with respect to the center of mass in x–z plane

\({I}_{5}\) :

Moment of inertia of leg with respect to joint1 in x–z plane

\({T}_{1}\) :

Torque of actuator1

\({T}_{2}\) :

Torque of actuator2

\({T}_{3}\) :

Torque of actuator3

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran

    S. Mohamad Hoseinifard

  2. Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran

    Majid Sadedel

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  1. S. Mohamad Hoseinifard
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Contributions

S.Mohamad Hoseinifard: Methodology, investigation, writing, analyzing, visualization, software.

Majid Sadedel: Supervision, project administration, conceptualization.

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Correspondence to Majid Sadedel.

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Hoseinifard, S.M., Sadedel, M. Analysis of Stability and Horizontal Motion of a Single Leg Hopping Robot. J Intell Robot Syst 108, 73 (2023). https://doi.org/10.1007/s10846-023-01864-9

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