*Jesse Haviland, Niko Sünderhauf, and Peter Corke*

IEEE Robotics and Automation Letters (RA-L), accepted January 2022.

We present the design and implementation of a taskable reactive mobile manipulation system. Contrary to related work, we treat the arm and base degrees of freedom as a holistic structure which greatly improves the speed and fluidity of the resulting motion. At the core of this approach is a robust and reactive motion controller which can achieve a desired end-effector pose while avoiding joint position and velocity limits, and ensuring the mobile manipulator is manoeuvrable throughout the trajectory. This can support sensor-based behaviours such as closed-loop visual grasping. As no planning is involved in our approach, the robot is never stationary *thinking* about what to do next. We show the versatility of our holistic motion controller by implementing a pick and place system using behaviour trees and demonstrate this task on a 9-degree-of-freedom mobile manipulator. Additionally, we provide an open-source implementation of our motion controller for both non-holonomic and omnidirectional mobile manipulators.

This approach can be used on both holonomic and omnidirectional mobile manipualtors. In the video below, we show it working on a 9 degree-of-freedom mobile manipualtor.

## How do I use it?

We have a created a robotics Python library called Robotics Toolbox for Python which allows our algorithm to be used an any robot. The following examples uses our Swift simulator.

Install Robotics Toolbox for Python and Swift using

```
pip3 install roboticstoolbox-python
```

### Position-Based Servoing Example on a non-holonomic mobile manipulator

```
import swift
import roboticstoolbox as rtb
import spatialgeometry as sg
import spatialmath as sm
import qpsolvers as qp
import numpy as np
import math
def step_robot(r: rtb.ERobot, Tep):
wTe = r.fkine(r.q)
eTep = np.linalg.inv(wTe) @ Tep
# Spatial error
et = np.sum(np.abs(eTep[:3, -1]))
# Gain term (lambda) for control minimisation
Y = 0.01
# Quadratic component of objective function
Q = np.eye(r.n + 6)
# Joint velocity component of Q
Q[: r.n, : r.n] *= Y
Q[:2, :2] *= 1.0 / et
# Slack component of Q
Q[r.n :, r.n :] = (1.0 / et) * np.eye(6)
v, _ = rtb.p_servo(wTe, Tep, 1.5)
v[3:] *= 1.3
# The equality contraints
Aeq = np.c_[r.jacobe(r.q), np.eye(6)]
beq = v.reshape((6,))
# The inequality constraints for joint limit avoidance
Ain = np.zeros((r.n + 6, r.n + 6))
bin = np.zeros(r.n + 6)
# The minimum angle (in radians) in which the joint is allowed to approach
# to its limit
ps = 0.1
# The influence angle (in radians) in which the velocity damper
# becomes active
pi = 0.9
# Form the joint limit velocity damper
Ain[: r.n, : r.n], bin[: r.n] = r.joint_velocity_damper(ps, pi, r.n)
# Linear component of objective function: the manipulability Jacobian
c = np.concatenate(
(np.zeros(2), -r.jacobm(start=r.links[4]).reshape((r.n - 2,)), np.zeros(6))
)
# Get base to face end-effector
kε = 0.5
bTe = r.fkine(r.q, include_base=False).A
θε = math.atan2(bTe[1, -1], bTe[0, -1])
ε = kε * θε
c[0] = -ε
# The lower and upper bounds on the joint velocity and slack variable
lb = -np.r_[r.qdlim[: r.n], 10 * np.ones(6)]
ub = np.r_[r.qdlim[: r.n], 10 * np.ones(6)]
# Solve for the joint velocities dq
qd = qp.solve_qp(Q, c, Ain, bin, Aeq, beq, lb=lb, ub=ub)
qd = qd[: r.n]
if et > 0.5:
qd *= 0.7 / et
else:
qd *= 1.4
if et < 0.02:
return True, qd
else:
return False, qd
env = swift.Swift()
env.launch(realtime=True)
ax_goal = sg.Axes(0.1)
env.add(ax_goal)
frankie = rtb.models.Frankie()
frankie.q = frankie.qr
env.add(frankie)
arrived = False
dt = 0.025
# Behind
env.set_camera_pose([-2, 3, 0.7], [-2, 0.0, 0.5])
wTep = frankie.fkine(frankie.q) * sm.SE3.Rz(np.pi)
wTep.A[:3, :3] = np.diag([-1, 1, -1])
wTep.A[0, -1] -= 4.0
wTep.A[2, -1] -= 0.25
ax_goal.T = wTep
env.step()
while not arrived:
arrived, frankie.qd = step_robot(frankie, wTep.A)
env.step(dt)
# Reset bases
base_new = frankie.fkine(frankie._q, end=frankie.links[2])
frankie._T = base_new.A
frankie.q[:2] = 0
env.hold()
```

### Position-Based Servoing Example on an omnidirectional mobile manipulator

```
import swift
import roboticstoolbox as rtb
import spatialgeometry as sg
import spatialmath as sm
import qpsolvers as qp
import numpy as np
import math
def step_robot(r: rtb.ERobot, Tep):
wTe = r.fkine(r.q)
eTep = np.linalg.inv(wTe) @ Tep
# Spatial error
et = np.sum(np.abs(eTep[:3, -1]))
# Gain term (lambda) for control minimisation
Y = 0.01
# Quadratic component of objective function
Q = np.eye(r.n + 6)
# Joint velocity component of Q
Q[: r.n, : r.n] *= Y
Q[:3, :3] *= 1.0 / et
# Slack component of Q
Q[r.n :, r.n :] = (1.0 / et) * np.eye(6)
v, _ = rtb.p_servo(wTe, Tep, 1.5)
v[3:] *= 1.3
# The equality contraints
Aeq = np.c_[r.jacobe(r.q), np.eye(6)]
beq = v.reshape((6,))
# The inequality constraints for joint limit avoidance
Ain = np.zeros((r.n + 6, r.n + 6))
bin = np.zeros(r.n + 6)
# The minimum angle (in radians) in which the joint is allowed to approach
# to its limit
ps = 0.1
# The influence angle (in radians) in which the velocity damper
# becomes active
pi = 0.9
# Form the joint limit velocity damper
Ain[: r.n, : r.n], bin[: r.n] = r.joint_velocity_damper(ps, pi, r.n)
# Linear component of objective function: the manipulability Jacobian
c = np.concatenate(
(np.zeros(3), -r.jacobm(start=r.links[5]).reshape((r.n - 3,)), np.zeros(6))
)
# Get base to face end-effector
kε = 0.5
bTe = r.fkine(r.q, include_base=False).A
θε = math.atan2(bTe[1, -1], bTe[0, -1])
ε = kε * θε
c[0] = -ε
# The lower and upper bounds on the joint velocity and slack variable
lb = -np.r_[r.qdlim[: r.n], 10 * np.ones(6)]
ub = np.r_[r.qdlim[: r.n], 10 * np.ones(6)]
# Solve for the joint velocities dq
qd = qp.solve_qp(Q, c, Ain, bin, Aeq, beq, lb=lb, ub=ub)
qd = qd[: r.n]
if et > 0.5:
qd *= 0.7 / et
else:
qd *= 1.4
if et < 0.02:
return True, qd
else:
return False, qd
env = swift.Swift()
env.launch(realtime=True)
ax_goal = sg.Axes(0.1)
env.add(ax_goal)
frankie = rtb.models.FrankieOmni()
frankie.q = frankie.qr
env.add(frankie)
arrived = False
dt = 0.025
# Behind
env.set_camera_pose([-2, 3, 0.7], [-2, 0.0, 0.5])
wTep = frankie.fkine(frankie.q) * sm.SE3.Rz(np.pi)
wTep.A[:3, :3] = np.diag([-1, 1, -1])
wTep.A[0, -1] -= 4.0
wTep.A[2, -1] -= 0.25
ax_goal.T = wTep
env.step()
while not arrived:
arrived, frankie.qd = step_robot(frankie, wTep.A)
env.step(dt)
# Reset bases
base_new = frankie.fkine(frankie._q, end=frankie.links[3]).A
frankie._T = base_new
frankie.q[:3] = 0
env.hold()
```

### Citation

If you use this work, please cite:

```
@article{haviland2022holistic,
author={J. {Haviland} and N. {Sünderhauf} and P. {Corke}},
journal={IEEE Robotics and Automation Letters},
title={A Holistic Approach to Reactive Mobile Manipulation},
year={2022},
volume={7},
number={2},
pages={3122-3129},
doi={10.1109/LRA.2022.3146554}}
}
```

### Acknowledgements

This research was supported by the Queensland University of Technology Centre for Robotics (QCR).