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AAS 15-627
A MOTION PLANNING METHOD FOR SPACECRAFT
ATTITUDE MANEUVERS USING SINGLE POLYNOMIALS
Albert Caubet
∗
, James D. Biggs
†
,
A motion planning technique for generating smooth attitude slew maneuvers
is presented, which can generate suboptimal feasible trajectories with low
computational cost in the presence of constraints. The attitude coordinates
are shaped by time-dependent polynomials, whose coefficients are deter-
mined by matching prescribed arbitrary boundary conditions. Quaternions
are used as the reference attitude parametrization for arbitrary maneuvers,
which require normalization of the four independently shaped coordinates.
In the case of spin-to-spin maneuvers, a particular combination of Euler
Angles are used. The torque profile is evaluated using inverse dynamics,
which allows the feasibility of the maneuver given the actuator constraints
to be checked. With this approach, a root-finding method is used to se-
lect the minimum time for a certain path. By increasing the degree of
the polynomial free coefficients are introduced, thus pointing constraints
can be accommodated and time can be optimized amongst this class of mo-
tion. This motion planning method is applied to a flexible spacecraft model,
demonstrating its effectiveness at reducing spillover vibrations.
INTRODUCTION
Motion planning refers to the problem of defining a feasible motion subject to constraints.
1
Additionally, the motion can be optimized according to a specified cost function. Polynomial
motion planning is widely used in robotics and computer graphics, for their efficiency and
ease of manipulation. Cubic splines are a popular method for connecting path points,
however, the acceleration is not smooth.
2
Additionally, obstacle avoidance usually requires
the computation of path points, which has been applied to spacecraft maneuvers in
3–5
with
various degrees of computational expense. In spacecraft slew maneuvers, motion planning
can be used to optimize a certain performance parameter and/or satisfy differential and path
constraints, before the trajectory is executed. Path or pointing constraints are exclusion
areas over the unit sphere which a certain body-fixed axis shall not enter, for instance
the boresight of some sensitive instruments must be kept at a minimum angle from the
Sun direction. Differential constraints include dynamic and kinematic constraints. They
generally require an internal model to calculate variables such as torque, in what is known
as an inverse dynamics approach. For instance, in a planned maneuver it must be ensured
that torque is not larger than the actuators’ limit at any point. Apart from torque, reaction
wheels experience saturation given by the wheels speed limit, which can also be assessed.
Sensors such as star trackers can have a maximum operational angular speed, and flexible
structures may require limits on acceleration and jerk to avoid excessive deflection and
∗
PhD candidate, Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
†
Associate Director, Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
1
vibration. In this paper, motion or trajectory planning is used to design the time evolution
of the attitude coordinates, while path planning refers to the representation of the attitude
in a time-independent space, such as the curves traced by the motion projected onto the
unit sphere.
In previous work on spacecraft attitude motion planning, McInnes
6
first shows the po-
tential of applying inverse dynamics to attitude maneuvers using polynomials and Euler
angles, while Biggs
7
finds reference motions for reorienting a spinning satellite by solving
analytically an optimal control problem, and Zhang
8
uses a 5
th
degree polynomial to obtain
a smooth eigenaxis rotation on a flexible spacecraft. In the field of computer graphics, Kim
9
proposes the use of exponential coordinates to satisfy the unit norm constraint of quater-
nions, with Tanygin
10
and Boyarko
11
applying this approach to spacecraft maneuvers using
inverse dynamics to minimize time and/or energy amongst this class of curves. Following
from the work by Boyarko,
11
Ventura
12
compares the performance (computational time and
optimization cost) of different attitude parameterizations using splines for motion planning,
and shows that the exponential functions in the quaternion parameterization require a larger
computational expense. The motion planning method proposed in this paper uses quater-
nions individually shaped by smooth polynomials, which are then normalized. Quaternions
are chosen since they are the preferred attitude coordinates for spacecraft applications due
to their efficiency and lack of singularities. Additionally, a special axis-azimuth parameter-
ization
13
is proposed for spin-to-spin maneuvers, in which the spinning axis is re-pointed,
due to their suitability to this specific problem.
Regarding optimization, time minimization amongst the class of polynomial trajecto-
ries is discussed. Other performance metrics are possible, such as energy (based on the
accumulated square root of torque) or fuel consumption. However, the planned trajecto-
ries are smooth, being more suited to reaction wheels than they are to a reaction control
system. With reaction wheels, fuel or energy optimization is not relevant (provided that
the spacecraft has enough power output to drive the wheels). Bilimoria and Wie
14
show
that the time-optimal maneuver has a bang-bang torque profile, which is not achievable
with the smooth torques provided by polynomial motion planning (although it can be close
enough with high order polynomials). Therefore, with this method the minimum time will
be selected within the set of feasible polynomial trajectories, which is limited by the maxi-
mum torque. Interestingly, Junkins
15
proves that for a single-axis rotational maneuver, the
energy-optimal trajectory is a polynomial, which suggests that this function family may
naturally provide energy-efficient maneuvers. In this paper, the time optimization strategy
employs a combination of root-finding and unconstrained optimization, aiming to mini-
mize computational cost and ensure convergence, as opposed to directly using nonlinear
programming solvers such as in
11
and.
12
Pointing constraints are also addressed, in combi-
nation with time optimization. In some works such as Frazzoli
16
and Tanygin,
17
pointing
constraints are satisfied using randomized path planning algorithms,
1
and then the time-
independent path is followed with a controller. In contrast, trajectory planning (as shaping
the attitude parameters with functions of time) addresses both path constraints and dy-
namic constraints by re-shaping the trajectory. Randomized path planning performs better
at finding a feasible geometric path in highly constrained spaces, but a priori it does not
consider the dynamic and kinematic aspects of the motion as trajectory planning does.
The smoothness of the resulting trajectories, along with the ability to monitor accelera-
2
tion and jerk, make this method particularly suited to spacecraft with flexible appendages.
Discontinuities in the torque profile such as in bang-bang maneuvers, or the initial step in-
put given by feedback controllers, result in infinite jerk leading to the excitation of flexible
modes and spillover (post-maneuver vibrations).
18, 19
Singh,
20
Kim,
21
and Byers
22
propose
solutions by smoothing the discontinuous torque switches of bang-bang maneuvers with a
variety of functions. The method in this paper aims to reduce vibrations with a guidance
approach rather than with a feedback control approach, by providing continuously smooth
torque along the trajectory—specially at the endpoints. In Singh
23
an input-shaped control
is used to minimize the tip deflection of appendages and spillover vibrations. The torque
profile obtained with the input-shaped method is made of discrete jumps (although not
bang-bang), but an interpolation of that discrete torque profile would result in a smooth,
sinusoidal-like shape. Interestingly, such a shape is similar to the torque profiles obtained
with inverse dynamics in this polynomial trajectory planning.
In motion planning, there is a trade-off between the pursuit of global optimality and com-
putational efficiency. The focus of this work is on computationally efficient motion planning
strategies, providing suboptimal results while respecting the constraints imposed on the at-
titude motion. The first section describes the method, detailing quaternion normalization,
the spin-to-spin case, and discussing the potential numerical issues associated with high-
order polynomials. The second section explores strategies to generate feasible trajectories in
terms of torque and path constraints, while finding the minimum possible maneuver time.
The final section applies the method to simulations with a flexible spacecraft, focusing on
the effects of smoothness regarding vibrations.
MOTION PLANNING METHOD OUTLINE
The proposed method represents the attitude of the body with a prescribed analytically
defined function of time. In this paper polynomial functions were chosen since they are
smooth, and easy to derive and manipulate. Polynomials are parameterized to match pre-
scribed boundary conditions on attitude, velocity, and higher order derivatives. Once the
desired attitude trajectory has been obtained, the torque profile can be obtained with in-
verse dynamics. Quaternions are used in this paper to parameterize attitude, as they are
non-singular and computationally efficient.
The trajectory of each quaternion in the S
3
unit sphere is shaped by the rational poly-
nomial function
q
i
(t) =
q
∗
i
(t)
kq
∗
(t)k
(1)
for i = 1, ..., 4, where q
∗
i
(t) is a polynomial:
q
∗
i
(t) = a
i0
+ a
i1
t + a
i2
t
2
+ ... + a
in
t
n
=
n
X
j=0
a
ij
t
j
(2)
Since these quaternions (depicted by the
∗
supercript) are individually shaped, they form
a vector in R
4
whose norm is not constant, thus each component i is normalized in Eq. (1)
using the quaternion unit norm
kq
∗
(t)k =
q
q
∗
1
(t)
2
+ q
∗
2
(t)
2
+ q
∗
3
(t)
2
+ q
∗
4
(t)
2
(3)
3
Arbitrary boundary conditions can be matched for any maneuver time t
f
. As the attitude
boundary conditions are normalized (i.e. q
i
(0) = q
∗
i
(0) and q
i
(t
f
) = q
∗
i
(t
f
) ) it is sufficient
to consider Eq. (2) to match the boundary conditions. This results in solving a simple linear
equation to find the value of the polynomial coefficients in Eq. (2). The m = n+1 boundary
conditions of the maneuver determine the degree of the polynomial and provide a system of
linear equations from which the coefficients a
ij
can be obtained, given a final maneuver time.
The minimum number of boundary conditions that define a slew maneuver are the initial
and final attitude and velocity (requiring a 3
rd
degree polynomial to define the quaternions’
trajectories). Additionally, acceleration boundary conditions can be introduced, with the
purpose of having zero torque at the trajectory endpoints. This may be required for flexible
spacecraft, to avoid vibration-inducing discontinuities in angular acceleration. For the same
reason, the boundary jerk (time derivative of acceleration) can be forced to zero so that it
is continuous at the endpoints.
The degree of the polynomial can be increased beyond n = m−1 (i.e. the minimum needed
for matching boundary conditions), which introduces degrees of freedom to the system in
the form of the additional coefficients. The extended polynomial of degree m − 1 + k, for
m boundary conditions and k additional terms, becomes
q
∗
i
(t) = a
i0
+ a
i1
t + a
i2
t
2
+ ... + a
i,m−1
t
m−1
+ ... + a
i,m−1+k
t
m−1+k
(4)
In this paper, a scenario with m = 8 boundary conditions is considered, matching attitude,
velocity, acceleration, and jerk. The 8 coefficients can thus be solved by a polynomial of
degree 7 representing q
∗
i
(t). However, we can increase the order of the polynomial with an
additional term (k = 1 in Eq. (4)). Assuming that the additional free coefficient a
i,8
is
guessed or known, the rest of the coefficients that make the trajectory match the boundary
conditions are determined by the following linear system of equations:
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 2 0 0 0 0 0
0 0 0 6 0 0 0 0
1 t
f
t
2
f
t
3
f
t
4
f
t
5
f
t
6
f
t
7
f
0 1 2t
f
3t
2
f
4t
3
f
5t
4
f
6t
5
f
7t
6
f
0 0 2 6t
f
12t
2
f
20t
3
f
30t
4
f
42t
5
f
0 0 0 6 24t
f
60t
2
f
120t
3
f
210t
4
f
a
i0
a
i1
a
i2
a
i3
a
i4
a
i5
a
i6
a
i7
=
q
i
(0)
˙q
i
(0)
¨q
i
(0)
...
q
i
(0)
q
i
(t
f
) − a
i,8
t
8
f
˙q
i
(t
f
) − 8a
i,8
t
7
f
¨q
i
(t
f
) − 56a
i,8
t
6
f
...
q
i
(t
f
) − 336a
i,8
t
5
f
(5)
where the vector on the right-hand side contains the selected boundary conditions. The
coefficients a
ij
of the i-th quaternion are then a function of the maneuver final time t
f
and
the corresponding boundary conditions.
The values of the additional coefficients can be selected in an optimization process, and
Eq. (5) is solved. In other words, the optimizer re-shapes the trajectory by adjusting the
additional coefficients, while maintaining the endpoints at the specified boundary condi-
tions. Alternatively, a more ”deterministic” approach can be considered, where k trajectory
waypoints (quaternions) are selected adding k equations to the system. In this case, besides
matching the boundary conditions, the trajectory will pass through the specified attitudes
at the specified times. At the waypoints, given the use of a single polynomial between
endpoints, the curve is smooth (i.e. of differentiability class C
∞
).
4
While Eq. (5) can be solved with linear algebra methods, it is more efficient to calculate
the coefficients using closed-form expressions (which can be rapidly obtained with a symbolic
mathematics software). Note that each polynomial degree has a different set of expressions.
The initial and final values of the quaternions’ time derivatives in Eq. (5) can be obtained,
given the boundary angular velocities, via the kinematics equation:
˙q
1
˙q
2
˙q
3
˙q
4
=
1
2
0 ω
3
−ω
2
ω
1
−ω
3
0 ω
1
ω
2
ω
2
−ω
1
0 ω
3
−ω
1
−ω
2
−ω
3
0
q
1
q
2
q
3
q
4
(6)
Furthermore, the boundary values of ¨q
i
and
...
q
i
are obtained by differentiating Eq. (6)
with respect to time. When selecting the endpoint attitudes, note that quaternions are
not unique in the sense that the same attitude in the SO(3) space can be represented
both by q and −q. However, a trajectory shaped between q
i
(0) and q
i
(t
f
) (t
f
being the
maneuver final time) is different than one connecting q
i
(0) with −q
i
(t
f
). This results in a
winding trajectory, where the desired attitude is reached through a long path. To avoid this
phenomenon, the sign of the quaternions should be selected according to a metric based
on the difference between q(0) and ±q(T
f
). Specifically, if q
d
is the difference between the
endpoint attitudes, expressed in quaternion algebra as
q
d
= q
f
· q
−1
0
(7)
the corresponding rotation angle θ
d
= 2 cos
−1
(q
d4
) should be less than 180 deg in order to
avoid a winding trajectory.
Spin-to-spin maneuvers
In the special case of spin-to-spin maneuvers, only the final pointing of a certain body
axis is relevant. Since quaternions define the full attitude (i.e. the three body axes), the
final spin phase angle must be determined, which adds another degree of freedom to the
problem that must be chosen. In order to avoid this, the direction of the pointing axis can be
parameterized with two spherical coordinates such as azimuth and declination angles (s
1
and
s
2
), which can be expressed independently as time polynomials. These two coordinates form
a reduced attitude parameterization,
24
while the phase angle of the other two orthogonal
vectors around the pointing axis is not specified. A third parameter (s
3
), describing the
rotation angle about the pointing axis, completes the full attitude in what is known as axis-
azimuth parameterization.
13
The s
3
is also expressed as a polynomial, however, its final
value s
f3
is not included in the boundary conditions set since it is irrelevant. This strategy is
particularly convenient for spin-to-spin maneuvers, i.e. transferring the spacecraft from one
pointing direction and spinning state
˙
s
0
= [0 0 ˙s
03
]
T
to another one with final
˙
s
f
= [0 0 ˙s
f3
]
T
.
In fact, the resulting attitude rotation formalism formed by s
1
, s
2
, and s
3
is a particular
combination of intrinsic Euler angles.
For instance, assume a spacecraft with an instrument aligned with the body axis y, which
is required to point in different directions. In this case, the attitude can be described as a
z −x
0
−y
00
Euler rotation, where the rotation matrix is R = R
z
(s
1
)R
x
(s
2
)R
y
(s
3
). The order
of the x and z rotations is not relevant, but the third rotation must be around the pointing
5
axis (y in this case). With 8 boundary conditions, the polynomials are of degree 7:
s
i
(t) = a
i0
+ a
i1
t + a
i2
t
2
+ ... + a
i7
t
7
(8)
For i = 1, 2. However, for i = 3, there are only 7 boundary conditions, thus a
37
= 0. The
coefficients are obtained by solving a linear system analogous to Eq. (5).
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
x
I
y
I
z
I
x
b
y
b
z
b
Figure 1. Path of the body axes on the unit sphere, in a spin-to-spin maneuver
0 100 200 300 400 500
−4
−2
0
2
4
time [s]
Euler angles [rad]
s
1
s
2
s
3
Figure 2. Trajectory of the attitude coordinates, in a spin-to-spin maneuver
Figure 1 shows the body axes paths of a spin-to-spin maneuver in an inertial frame, where
the pointing axis y
b
precesses towards the target direction (depicted by a point at the end of
the path line) while the other two orthogonal axes keep rotating about it. In Figure 2, the
two coordinates defining the direction of the pointing axis (s
1
and s
2
) are driven to their
final desired values, while s
3
follows a constant rate trajectory (since the prescribed initial
and final spin rates are the same) where the final value of the angle is not relevant.
The singularity associated with Euler angles occurs when calculating their time derivatives
with the kinematics equation at angles of 90 deg. However, in this case the kinematic
equation is not used, since the time derivatives of the angles are obtained by differentiating
the polynomial. Also, no singularities arise when evaluating the angular velocities and
accelerations (needed to calculate the torque). Finally, the trajectory can be converted to
6
quaternions if required by the attitude control system of the spacecraft, at the cost of having
to use trigonometric functions.
Numerical stability of high degree polynomials
High degree polynomials may have sensitivity issues, where small errors in the inputs
cause relatively large errors in the outputs. In the attitude control scenario, the main
source of error comes from sensor inaccuracies, namely the current attitude and velocity
values that are inputs to the linear system. Mathematically speaking, the matrix of the
linear system in Eq. (5), expressed in the form Ax = b, can be ill-conditioned for too high
polynomial orders and t
f
values. A matrix is ill-conditioned if it is close to being singular,
therefore non-invertible. A metric of the conditioning of a matrix is its condition number
C, or the ratio of the largest to smallest singular value in the singular value decomposition.
The order of magnitude of C gives an estimate of the digits of accuracy lost in solving a
linear system with that matrix.
In order to reduce the condition number of the matrix, the time domain can be scaled so
that the final time is 1, to prevent some elements in A from being too large. For instance,
with the new variable τ ∈ [0, 1], where τ = t/t
f
, the condition number is reduced from
10
18
(for m = 8 and t
f
= 300 sec) to 10
4
. The scaled coefficients can be calculated by
solving the corresponding linear system, but the vector of boundary values changes due to
the differentiation with respect to a scaled variable. The differential operator with respect
to time can be expressed as
d
dt
=
d
dt
(τ)
d
dτ
(9)
by replacing τ = t/t
f
, Eq. (9) becomes
d
dt
=
1
t
f
d
dτ
(10)
which can be raised to the k-th derivative. Thus, the corresponding k-th time derivatives
of the scaled quaternions are calculated as
d
k
q
i
(τ)
dτ
k
= t
k
f
·
d
k
q
i
(t)
dt
k
(11)
and the boundary values in the right-hand side vector of Eq. (5) must be adjusted accord-
ingly (while in the matrix, t
f
= 1). The scaled coefficients a
in
of the polynomial in τ are
related to the original ones by
a
in
=
a
in
t
n
f
(12)
MINIMUM TIME MANEUVERS IN A CONSTRAINED SPACE
The maneuver time t
f
and any free coefficients of the polynomial must be selected in
order to calculate the rest of the coefficients, defining a trajectory matching the bound-
ary conditions. Those variables can be selected in an optimization process, so that the
minimum t
f
is chosen amongst the set of feasible polynomial trajectories. In this section,
the constraints considered are torque, which is evaluated from the quaternions and their
derivatives with an internal model (inverse dynamics), and pointing keep-out areas. While
7
this is a constrained optimization problem, the use of sequential quadratic programming
(SQP) algorithms has been avoided. Instead, in a more efficient approach, a combination
root-finding and unconstrained optimization has been used.
Inverse dynamics
A simple model of a fully actuated rigid body has been used to obtain the torque profile.
The Euler’s equation of rigid-body dynamics relates the torque u
i
(along the body i-th axis)
to the angular velocity ω
i
and acceleration ˙ω
i
and principal moments of inertia I
i
, as
u
1
= I
1
˙ω
1
− (I
2
− I
3
)ω
2
ω
3
u
2
= I
2
˙ω
2
− (I
3
− I
1
)ω
1
ω
3
u
3
= I
3
˙ω
3
− (I
1
− I
2
)ω
1
ω
2
(13)
The angular velocities and accelerations are related to quaternions and their time derivatives
through the rotational kinematics,
25
as
ω
1
= 2( ˙q
1
q
4
+ ˙q
2
q
3
− ˙q
3
q
2
− ˙q
4
q
1
)
ω
2
= 2( ˙q
2
q
4
+ ˙q
3
q
1
− ˙q
1
q
3
− ˙q
4
q
2
)
ω
3
= 2( ˙q
3
q
4
+ ˙q
1
q
2
− ˙q
2
q
1
− ˙q
4
q
3
)
(14)
˙ω
1
= 2(¨q
1
q
4
+ ¨q
2
q
3
− ¨q
3
q
2
− ¨q
4
q
1
)
˙ω
2
= 2(¨q
2
q
4
+ ¨q
3
q
1
− ¨q
1
q
3
− ¨q
4
q
2
)
˙ω
3
= 2(¨q
3
q
4
+ ¨q
1
q
2
− ¨q
2
q
1
− ¨q
4
q
3
)
(15)
Analytical expressions for the quaternion derivatives are obtained by differentiating Eq.(1)
with respect to time:
˙q
i
(t) =
˙q
∗
i
(t)
kq
∗
(t)k
−
q
∗
i
(t)
kq
∗
(t)k
3
4
X
i=1
q
∗
i
(t) ˙q
∗
i
(t)
!
(16a)
¨q
i
(t) =
¨q
∗
i
(t)
kq
∗
(t)k
−
˙q
∗
i
(t)
kq
∗
(t)k
3
4
X
i=1
q
∗
i
(t) ˙q
∗
i
(t)
!
+
q
∗
i
(t)
3
kq
∗
(t)k
5
4
X
i=1
q
∗
i
(t) ˙q
∗
i
(t)
!
2
−
1
kq
∗
(t)k
3
4
X
i=1
( ˙q
∗
i
(t)
2
+ q
∗
i
(t)¨q
∗
i
(t))
!
(16b)
Since the quaternions in Eqs. (14) and (15) can be replaced by their corresponding time-
dependent polynomials (Eq. (1) and their derivatives), whose coefficients are a function of
the maneuver time t
f
, ultimately the torque is a function of t and t
f
(given a set of bound-
ary conditions for a particular maneuver). Evaluating the torque along the trajectory is
essential to ensure that the actuators always remain within their operative limits. Simi-
larly, an additional time differentiation of Eq. (15) allows for the evaluation of jerk along
the maneuver.
If the actuators are reaction wheels, it can be useful to assess the speed buildup during
the maneuver, to ensure that they will not become saturated. Assuming that the wheels
are aligned with the body axes, the planned torque can be related to the derivative of their
angular momentum. The body angular velocity is considered negligible compared to the
8
magnitude of typical wheels’ speeds. Therefore the wheel’s acceleration and moment of
inertia can be related to the torque provided along its axis by
u ≈ −I
W
˙
ω
W
(17)
Where the vector
˙
ω
W
contains the wheels’ angular acceleration and I
W
= diag(I
W 1
, I
W 2
, I
W 3
)
is their inertia matrix. The wheel speeds are obtained by replacing Eq. (13) into Eq. (17)
and integrating:
ω
W
(t) ≈ −
1
I
W
I(ω(t) − ω(0)) +
Z
t
0
ω × (Iω)dt
+ ω
W
(0) (18)
where the angular velocity of the body ω can in turn be replaced by Eq. (14). While
the integral in Eq. (18) has a closed-form solution as a function of time and the polyno-
mial coefficients, it is so complex that it is computationally more efficient to evaluate it
numerically.
Maneuver time minimization
The final time t
f
is required to evaluate the coefficients in Eq. (5). In this section the
goal of finding a minimum t
f
is addressed, which requires the evaluation of the torque. The
maneuver duration affects the torque profile, with shorter final times resulting in higher
torques. A criterion for choosing t
f
is to find its minimum value such that the calculated
maximum torque in the maneuver (of any axis, in absolute value) is equal to the actuator’s
torque limit u
lim
. In an analogous way, other differential constraints can be considered,
such as a limit on velocity, acceleration, jerk, or the reaction wheels rate.
It is possible to obtain an expression of the planned torque as a function of time and t
f
by
combining the dynamic equations and the polynomials representing the attitude parameters,
as mentioned above. However, due to the high non-linearity of this expression, finding the
minimum t
f
with a purely analytical approach is not practical. A more efficient strategy
consists in discretizing the trajectory and evaluating the torques at each node (u
i
(t
k
) for
the i-th axis and k-th node). The difference with the torque limit u
lim
is calculated for each
node and the maximum value of the set is obtained:
J
i
= max
k
(|u
i
(t
k
)| − u
lim
) (19)
where J
i
is the largest difference amongst all nodes of the i-th torque profile. The three
axes can be combined in J = max {J
1
, J
2
, J
3
}.
As shown in Figure 3, the optimum point corresponds to J = 0, whereas if J > 0, the
maximum torque is above the limit. While gradient-based optimization algorithms would
use J
2
to find the optima, this performance index allows to (a) know if a trajectory is feasible
(in terms of torque) simply by checking the sign of J, and (b) use a root-finding algorithm,
which are more efficient than numerical gradient-based ones. For general maneuvers with
arbitrary boundary angular velocities, accelerations, and jerks, the shape of J as a function
of t
f
may feature multiple local optima—some of which could be unfeasible. Thus, an
algorithm capable of finding the global optimum would be required to compute the minimum
feasible t
f
. However, in the cases of rest-to-rest and spin-to-spin maneuvers, the evolution
9
Figure 3. Illustration of torque profiles with different final times, with
J being a metric of the peak height
0 500 1000 1500 2000 2500 3000 3500 4000
−1
−0.5
0
0.5
1
1.5
2
T
f
[s]
J
spin−to−spin
rest−to−rest
Figure 4. Evolution of J (maximum torque over the limit) with maneuver time
of J with t
f
is monotonically decreasing, as shown in the example of Figure 4. In this case,
finding the root of this curve, corresponding to the minimum t
f
of that particular path, is
performed with very few iterations.
However, other paths (still matching the prescribed boundary values) may have a lower
final time, which can be explored by polynomials with additional terms. In this case, free
variables are added to the system, whose value can be selected by an optimizer. Best results
are obtained with a two-layer optimization approach. The outer layer explores the space of
4×k free coefficients (considering 4 quaternions, shaped with polynomials of degree m + k),
using an unconstrained optimization method with maneuver time as the performance index.
At every iteration of the optimizer, the root-finding algorithm runs as the inner loop, giving
the minimum feasible time for that specific trajectory. This approach is robust in the sense
that, even if the optimizer converges to a local optimum, the trajectory will satisfy torque
constraints. While there exists a global minimum time within the set of trajectories given
by polynomials of a certain degree, this value can be reduced if higher degrees are used.
The absolute global optimum of the problem could be that given by an optimal control
computational method (although optimality is not guaranteed either).
10
Obstacle avoidance
With the additional degrees of freedom provided by free variables in the polynomial,
the path of the body axes on the unit sphere can be diverted in order to avoid pointing
constraints. The static path constraints
3
or keep-out areas are represented by cones inter-
secting the unit sphere. The resulting circle should not be trespassed by the path of the
corresponding body axis i, in other words, the angle between the body axis v
I
i
(t) (resolved
in the inertial frame) and the cone axis w
c
should not be lower than the cone angle γ
c
:
v
I
i
(t) · w
c
≤ cos(γ
c
) (20)
The pointing body axis can be drawn from quaternions with
v
I
i
= (q
2
4
− k~qk
2
)v
B
i
+ 2(~q
T
v
B
i
)~q + 2q
4
(~q × v
B
i
) (21)
where ~q = [q
1
, q
2
, q
3
]
T
and v
B
i
is the pointing axis resolved in the body frame.
Obstacle avoidance is carried out using unconstrained optimization of the free parameters.
Constraints can be considered in unconstrained optimization with a performance index
known as penalty function. The penalty function is applied to the keep-out area:
J
OA
=
max
k
(v
I
i
(t
k
) · w
c
− cos(γ
c
)) if v
I
i
(t) · w
c
> cos(γ
c
);
0 otherwise.
(22)
where the trajectory is discretized, with v
I
i
(t
k
) depicting the k-th node. Essentially, the
penalty function in Eq. (22) is related to the closest distance to the center of the cone
(as far as the resolution of the discretization allows) when the trajectory path crosses it.
During the optimization, the path ”slides down” out of the keep-out area. When the path
of the axis is tangent or anywhere outside the keep-out area, the penalty is constant at
zero—deactivating the obstacle avoidance process.
Figure 5. Two-layer time optimization algorithm
The obstacle avoidance penalty function can be included in the two-layer optimization
for finding the minimum time described in the previous section. As soon as the path crosses
11
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
y
x
z
I
Figure 6. Paths of the pointing body axis on the unit sphere and keep-
out circle, for the nominal trajectory (grey line) and the diverted one
(black line)
a keep-out area, the obstacle avoidance algorithm is activated and time minimization is not
considered, until the penalty index J
OA
becomes zero. The flow chart of the algorithm of
time optimization with pointing constraints is shown in Figure 5. The torque constraints
are embedded in the inner loop (the root-finding process described in the previous section),
which are used to find the minimum t
f
for every iterated combination of free coefficients.
This t
f
becomes the performance index if obstacle avoidance is not activated (J
OA
= 0).
An interesting property of rest-to-rest maneuvers, for some given values of free coefficients,
is that the axes’ paths remain unchanged when varying the maneuver time t
f
. Therefore
in the obstacle avoidance part of the algorithm, the final time can be set to t
f
= 1.
Figure 6 shows the path of the pointing axis (black line), using a single additional term
on the polynomials (k = 1 in Eq. (4))—therefore, four free variables are included. The grey
line shows an initial-guess trajectory (minimum polynomial degree for matching boundary
conditions) which does not satisfy the path constraint. While the obstacle avoidance al-
gorithm forces the path out of the cone, in this case the one featuring the minimum time
happens to cross the keep-out area, making the optimizer converge to a path tangent to the
circle (i.e. the minimum-time trajectory satisfying the pointing constraint).
Alternatively, obstacle avoidance can be achieved in a deterministic way using a path
point (thus adding an additional equation to the linear system in Eq. (5)). The point is
selected by calculating the nearest point of the nominal path to the cone center and moving
it to the closest point on the circle. This implies rotating the attitude about the axis
w
c
× v
I
i
(τ
k
) by an angle γ
c
− γ. Since time is adimensional, the τ
k
associated with the new
point is the same as the original one. The resulting path of the pointing axis is tangent
to the keep-out cone. This approach is suboptimal, but as it is deterministic it avoids the
unconstrained optimization process and a feasible motion is obtained.
12
Note that on maneuvers with arbitrary endpoint velocities, the variable t
f
affects the
path, thus the time-minimisation and the obstacle avoidance problems are coupled. The
two-layer approach can still be used, but in this case the actual t
f
must be considered in the
outer layer (where path constraints are checked). However, in the case of spin-to-spin using
the previously described parameterization, the path of the pointing axis is not dependent
on t
f
, and an analogous approach to the rest-to-rest scenario can be used.
SIMULATIONS ON A FLEXIBLE SPACECRAFT
So-called flexible spacecraft are non-rigid bodies, which may have flexible appendages
and/or multiple parts with non-rigid links. Flexibility introduces additional challenges to
attitude maneuvering due to oscillations of the appendages and a varying inertia matrix.
To study how the proposed method performs in this scenario, simulations on a flexible
spacecraft with reaction wheels were realized. The spacecraft has been modeled as a multi-
body object, formed by a central cuboid with two solar panels, joint together by hinges
(torsional spring + damper). The model used in the simulations is detailed in Shahriari.
26
As opposed to modeling a spacecraft with Bernoulli beams,
15
which require assumed modes
of vibration, a multi-body only needs the stiffness and damping rate of the joints to be
specified. Also, as shown in Shahriari,
26
it is shown to be more representative of the three-
dimensional case when compared to a finite elements model.
C.M.
C
e
n
θ
e
l
e
r
b
1
b
2
b
3
Figure 7. Spacecraft with hinged symmetric panels
Once the polynomial coefficients are obtained for a particular maneuver in the motion
planning process, the desired state and predicted torque are computed at each time step.
Due to the torque being planned with a simplified, rigid-body internal model, if inputted in
open loop there will be an error in the attitude profile if flexible structures or other major
disturbances are involved. Therefore, a proportional-derivative (PD) quaternion feedback
controller
25
was implemented, which tracks the desired attitude and angular velocity. Since
the planned torque profile is available, the controller is augmented by including it as a
feed-forward command. The control law is
u
cmd
= −K
p
q
e
− K
d
ω
e
+ u
pln
(23)
where K
p
and K
d
are the proportional and derivative coefficients, q
e
is the error quaternion,
ω
e
is the error angular velocity, and u
pln
is the torque as obtained in the planning.
The simulations have been realized considering a 2000-kg spacecraft, with a body inertia
of I
b
= diag(310, 310, 310) kgm
2
and 3×1 m solar panels with a mass of 40 kg. The hinges
linking the panels with the body have a stiffness of 12 N m rad
−1
and a damping of 0.01
kg m
2
s
−1
rad
−1
. The simulated maneuvers have been performed considering reaction wheels
limited to 0.2 Nm.
Figure 8 shows the quaternions of a rest-to-rest maneuver, while Figure 9 shows the
torque profile. Solid lines are the actual states, whereas dashed lines represent the planned
13
0 50 100 150 200 250 300
−1
−0.5
0
0.5
1
time [s]
attitude
q
1
q
2
q
3
q
4
Figure 8. Quaternions trajectory
0 50 100 150 200 250 300
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
time [s]
torque [Nm]
u
1
u
2
u
3
Figure 9. Torque profile
trajectory. Note that whereas quaternions are tracked with precision, the actual torque
(as given by the controller) is higher than the planned profile due to disturbance rejection
caused by large flexibility. This result emphasizes the need for applying an appropriate
safety margin during the motion planning phase, to avoid the actual torque from reaching
the actuators’ limit.
0 50 100 150 200 250 300 350 400
−0.02
0
0.02
0.04
0.06
0.08
0.1
time [s]
θ
p
[rad]
poly. deg. 3
poly. deg. 5
poly. deg. 7
Figure 10. Solar panel deflection angle
Figures 10 and 11 show, for the rest-to-rest case, the evolution of the deflection angle
of the solar panels at the hinges θ
p
, which relates to the panels’ tip displacement, and its
derivative
˙
θ
p
. Polynomials of different degrees were tested. Zero jerk at the boundaries or
endpoints of the trajectory can be achieved by a trajectory shaped with a polynomial of
degree 7, zero acceleration can be achieved by a polynomial of degree 5, and a polynomial
14
0 50 100 150 200 250 300 350 400
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10
−3
time [s]
˙
θ
p
[rad/s]
poly. deg. 3
poly. deg. 5
poly. deg. 7
Figure 11. Time derivative of the panel deflection angle
of degree 3 suffices to enforce zero velocity. The initial and final torques planned with a
3
rd
degree polynomial are not zero, thus the jerk is infinite causing short-period vibration
modes (which is better appreciated in
˙
θ
p
). However, the feedback control tracking the
trajectory acts as a damper of those initial oscillations, that would remain if open-loop
control was used. When the torque is cut off the spacecraft is left with residual vibrations
on the spacecraft, an issue known as spillover. This situation is significantly improved with
a polynomial of degree 5. The initial and final jerk, while not zero, are limited, significantly
reducing the initial vibration and the spillover. With this level of stiffness at the hinges, the
initial vibrations are barely noticeable—and quickly damped by the feedback control. With
a 7
th
degree polynomial enforcing zero jerk, virtually no initial short-period vibrations are
generated. A small yet significantly reduced spillover can still be observed in the simulations,
caused by the offset between the planned torque and the actual control torque.
Taking the aforementioned into consideration, choosing the right polynomial for shap-
ing the trajectory will depend mainly on the specific mission requirements and constraints.
Since the maneuver time is fixed, lower degree polynomials plan trajectories with lower
acceleration and torque peaks, leading to smaller maximum tip displacement. Thus, given
the same torque limit, lower degree polynomials can plan the same maneuver with a smaller
final time, at the expense of being more aggressive at the endpoints. The smoothest yet
longest maneuvers are achieved using a polynomial of degree 7, whereas time can be slightly
reduced by using a trajectory allowing non-zero initial jerk as represented by a 6
th
degree
polynomial (the closed-loop feedback controller will damp the initial vibrations). It is ad-
vised to use a very smooth ending torque to prevent spillover, as well as to implement
trajectory tracking rather than open-loop control—to both minimize the attitude error and
damping vibrations. If there is a constraint on some appendage’s maximum tip displace-
ment, a limit on maximum acceleration can be considered so that a higher t
f
may be chosen,
even if the torque is well below the actuators’ limit. Finally, with this method, the design
of the spacecraft structure can be allowed to be less stiff, and still prevent large oscillations.
CONCLUSIONS
An inverse dynamics trajectory planning method using polynomials has been presented.
The polynomials shape the quaternions of an attitude maneuver by matching arbitrary
boundary conditions on attitude, velocity, and higher derivatives (acceleration, jerk) if re-
quired. The torque profile of the actuators can be assessed, allowing torque limits to be
15
satisfied. By increasing the degree of the polynomial there are more coefficients than those
required to match boundary conditions. Those extra free coefficients can be used as op-
timization variables to further minimize time (or any other cost function) and deal with
pointing constraints. A combination of root-finding and unconstrained optimization us-
ing few variables (four free coefficients plus the final time, in the examples of this study)
has been proposed, avoiding the more computationally expensive nonlinear programming
solvers. With this approach, suboptimal but feasible trajectories are found in a constrained
space at low computational cost. The limits of this method arise in highly constrained
spaces, where there may not be enough degrees of freedom to find a feasible trajectory.
Apart from trajectory constraints such as torque limits and keep-out areas, the ability to
generate smooth motions and enforce boundary conditions on acceleration and jerk has
proven relevant to flexible spacecraft, specially for the reduction of spillover vibrations.
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