IEEE Robotics & Automation Magazine - March 2014 - 67

2.5

1.5

sf (rad)

1.5

2.5

2
2

2.5

1
-2

-1.5 -1

2.5

1.5
-0.5

0
0.5
k1(rad)
(b)

1.5

y-Constraint Error (m) Versus Parameterized
Freedom Values (k1, s1)
3
2

2.5

1
-2

1.5

2.5

1.5

0.5

1.5

0.5

1.5

2.5

-0.5

2.5

-1

sf (rad)

-1

0.5

3.5

The Jacobian of this state prediction integral is, once again,
computed numerically. For offline use, multiple observations
can be used to generate an overdetermined system of equations to be solved for the disturbance parameters. A more elegant method would be formulating an identification Kalman
filter, which runs continuously while the system operates.

3.5

4.5

-1.5 -1

-0.5

.5
-0
-1

Local Search Graphs
Local motion planning and obstacle avoidance is the problem of search in the space of horizon-limited actions to select a motion that keeps a mobile robot safe and moving in
a way that satisfies some objectives. The planning horizon
can be described using distance or time, but it should extend to or beyond the stopping distance of the platform in
the local environment. The predictive motion model used
for the local motion planner must also reasonably approximate the vehicle response to a set of actions because feedback control can, in certain situations, execute maneuvers

2.5

3.5

(5)

0.5

2.5

The disturbances can be visualized as wheel slip, but because all prediction error is being calibrated, this means that
other sources of error are being converted to equivalent wheel
slip. All velocities in the state are expressed in body coordinates so that they are constant under steady state turning conditions. In the systematic model, the task is to predict the
system state at future time t 1

#t t xo ^x, u + du (p, x, u)hdt.

-0
0

4.5

2.5
2
1.5

= p 1 V + p 2 ~ + p 3 V~ + p 4 sin (z) + g. (4)

x ^t 1h =

x-Constraint Error (m) Versus Parameterized
Freedom Values (k1, s1)
0.5

du (p, x, u)

(a)

1.5

motion model can be used either online or offline, by observing its integrated effect over short time intervals, and
comparing that result to direct measurements [6].
The reader may wonder why a model is needed if measurements are available. There are two reasons, both derived
from the fact that a predictive model is desired that predicts
motion from commands, rather than estimating it from
measurements. When calibrating online, the measurements
are not available until after the motion has been executed.
The process of continuously comparing predictions with
measurements refines and adapts the model. Conversely,
when calibrating offline, the measurements can be generated
based on high-performance ground-truth sensing, perhaps
based on infrastructure, which would not be available during
normal operation.
While many alternatives exist, we have used a formulation
that associates all prediction error with presumed disturbance
inputs. We assume that the disturbances du (p, x, u) depend
on some parameters and the state and inputs u. For example,
the commands may be linear V and angular velocity ~ . The
attitude of the vehicle (z, i), or equivalently the gravity vector,
is also considered to be an aspect of the state. Linear and quadratic terms are included in the model. In the following example expression, the V~ term captures the effect of lateral
acceleration:

0
0.5
k1(rad)
(c)

1.5

Figure 3. An example of model-predictive trajectory generation
applied to motion planning in rough terrain for a planetary rover
with a rocker-bogie chassis: (a) shows the generated trajectory
that satisfies the terminal state, initial state, and motion model
constraints, and (b) and (c) show contour plots of the constraint
error with respect to parameterized freedom values and an
overlay of the history of parameter corrections by the trajectory
generation algorithm.

MARCH 2014

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2014