# Optimal Whole Body Trajectory Planning for Mobile Manipulators in Planetary
Exploration and Construction

Federica Storiale1, Enrico Ferrentino1, Federico Salvioli2, Konstantinos
Kapellos3 and Pasquale Chiacchio1 1Authors are with the Department of
Information and Electrical Engineering and Applied Mathematics (DIEM),
University of Salerno, Fisciano, SA, 84084, Italy, {fstoriale,eferrentino,
<EMAIL_ADDRESS>Salvioli is with Robotic Exploration, ALTEC
S.p.A., 10146 Torino, Italy<EMAIL_ADDRESS>Kapellos is with Trasys International, 3001 Heverlée, Belgium,
<EMAIL_ADDRESS>
###### Abstract

Space robotics poses unique challenges arising from the limitation of energy
and computational resources, and the complexity of the environment and
employed platforms. At the control center, offline motion planning is
fundamental in the computation of optimized trajectories accounting for the
system’s constraints. Smooth movements, collision and forbidden areas
avoidance, target visibility and energy consumption are all important factors
to consider to be able to generate feasible and optimal plans. When mobile
manipulators (terrestrial, aerial) are employed, the base and the arm
movements are often separately planned, ultimately resulting in sub-optimal
solutions. We propose an Optimal Whole Body Planner (OptiWB) based on Discrete
Dynamic Programming (DDP) and optimal interpolation. Kinematic redundancy is
exploited for collision and forbidden areas avoidance, and to improve target
illumination and visibility from onboard cameras. The planner, implemented in
ROS (Robot Operating System), interfaces 3DROCS, a mission planner used in
several programs of the European Space Agency (ESA) to support planetary
exploration surface missions and part of the ExoMars Rover’s planning
software. The proposed approach is exercised on a simplified version of the
Analog-1 Interact rover by ESA, a 7-DOFs robotic arm mounted on a four wheels
non-holonomic platform.

##  I Introduction

In the context of planetary colonization, space robots act as human precursors
for exploration, terrain preparation and infrastructures construction [1],
performing activities otherwise unfeasible for men, allowing a reduction of
risks and costs. Common scenarios include robotic arms mounted on rovers, that
manipulate, transport, and assemble components for building infrastructures on
the Moon or Mars, or on spacecrafts, for samples collection [2]. These
applications require robots to move safely and efficiently in a challenging
environment. Because of limited resources, special attention must be paid to
energy and computational requirements. In this context, offline trajectory
planning, i.e. on ground, where the unit cost of computational power is lower,
plays a fundamental role both in the mission design phase, for feasibility and
budget assessments, and in the operations phase, to deliver efficient activity
plans. Offline planning allows engineers to tailor trajectories for each task
according to optimality criteria, taking into account the robot’s and the
environment’s constraints. For example, it ensures that the robot moves
smoothly, accurately, and efficiently to the desired location, while avoiding
collisions and non-traversable areas. This is essential to prevent damage to
the robot and to ensure the integrity of collected samples and manipulated
objects.

In space robotics, the planning problem is often decomposed: the trajectory
for the mobile base is planned through sampling-based approaches [3, 4, 5, 6]
or dynamic programming-inspired algorithms [7], while the arm motion is
computed through inverse kinematics, spline interpolation, genetic algorithms,
reachability analysis [8]. The reason behind the choice of a decoupled
approach is the increased complexity of whole-body planning, also in terms of
computational resources [7], as it requires managing multiple degrees of
freedom and different constraints for the arm and the base at the same time.
However, this separation can only generate sub-optimal plans [9] and, in the
worst case, compromise feasibility [8]. On the contrary, whole-body motion
planning [10, 11, 12, 13] allows exploiting the system’s kinematic redundancy
to accomplish secondary objectives, optimizing performance indices of
interest. Discrete Dynamic Programming (DDP) is a global optimization
technique that is frequently employed when an efficient solution is needed to
a non-convex trajectory planning problem [14, 15, 16]: it has been proven
effective for mobile manipulators [12] and constrained systems in general
[17]. Nevertheless, its discrete nature yields non-smooth trajectories, whose
conversion into actual commands for real systems remains questionable [12].

In this paper, building upon the algorithm proposed in [12], we design an
Optimal Whole Body Planner (OptiWB) preserving solution optimality through DDP
and guaranteeing trajectory smoothness through optimal interpolation. We
demonstrate the effectiveness of our approach in a real space exploration use
case, where kinematic redundancy is exploited to improve camera visibility
over a target that must be reached by the robotic arm. The whole-body approach
of OptiWB allows generating efficient plans, while simplifying the planning
process at the control center: the ground operator can generate complex
synchronized movements of the mobile platform and the arm by solely providing
the desired trajectory for the arm’s end-effector. This procedure is faster
and less error-prone than defining cumbersome maneuvers for the rover and the
arm independently.

While OptiWB is implemented in ROS, which would make its future integration
with Space ROS [18] effortless, we demonstrate the versatility of our planner
by integrating it with the 3D Rover Operations Control Software (3DROCS) [19,
20], a commercial mission planning system by Trasys International, already
employed in missions of space exploration and colonization. The robotic
platform is a simplified version of the Analog-1 Interact rover [21], composed
of a differential drive non-holonomic mobile base and a LBR iiwa 14 R820 7-DOF
arm [22].

In Section II, OptiWB is presented, with a description of both the DDP-based
planner and the optimal interpolation. In Section III and Section IV, the
constraints and the objective function designed for the considered exploration
use case are described, respectively. The integration with 3DROCS is discussed
in Section V. The results of applying OptiWB to our use case are presented in
Section VI, while Section VII draws the conclusions of our work.

##  II Optimal Whole Body Planner (OptiWB)

The high-level architecture of OptiWB is given in Fig. 1. Each component is
described next.

Figure 1: Optimal Whole-Body Planner (OptiWB) architecture.

###  II-A Discrete Dynamic Programming

Let ${\bf x}(t):\mathbb{R}\rightarrow\mathcal{X}\subseteq SE(3)$ be the
desired task space trajectory with respect to time $t$, designed at the ground
control center, either as a result of a higher-level task planner, or
manually, by human operators, possibly by the means of graphical tools. Every
${\bf x}(t)$, for a fixed $t$, is termed a _pose_ and is characterized by $m$
degrees of freedom, which is the dimension of $\mathcal{X}$. The joint space
commands (or joint space trajectory) ${\bf
q}(t):\mathbb{R}\rightarrow\mathbb{R}^{n}$ for the robot are related to the
task space trajectory, at the positional (zero-order) level, through inverse
kinematics, i.e.

${\bf q}(t)={\bf k}^{-1}\big{(}{\bf x}(t)\big{)},$ (1)

where ${\bf k}:\mathbb{R}^{n}\rightarrow SE(3)$ is the forward kinematics
function. When the number of joints $n$ is greater than the number of
variables needed to describe the task $m$, the degree of redundancy is
$r=n-m>0$. For redundant robots, (1) admits an infinite number of solutions,
that are joint configurations corresponding to the same task. The process of
solving (1) for each $t$, when $r>0$, is termed redundancy resolution and can
be performed by optimizing some performance criterion.

The approach proposed in [17] is based on redundancy parametrization [23],
i.e. the choice of a set of joint positions (or functions of them)
$\boldsymbol{\nu}\in\mathbb{R}^{r}$ representing redundancy, such that (1)
becomes:

${\bf q}(t)={\bf k}_{\alpha}^{-1}\big{(}{\bf
x}(t),\boldsymbol{\nu}(t)\big{)},$ (2)

where ${\bf k}_{\alpha}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is the
forward kinematics ${\bf k}$, augmented with the redundancy parameters’
equations. The mapping in (2) is square, hence, for fixed $\boldsymbol{\nu}$
and ${\bf x}$, the vector of joint positions ${\bf q}$ belongs to a finite
set.

Considering systems composed of a mobile base and an arm, ${\bf q}=\left[{\bf
q}_{b}^{T},{\bf q}_{a}^{T}\right]^{T}$, where ${\bf
q}_{b}\in\mathbb{R}^{n_{b}}$ is the base configuration, ${\bf
q}_{a}\in\mathbb{R}^{n_{a}}$ is the arm configuration, and $n_{a}+n_{b}=n$.
For terrestrial robots, for the sake of planning, we take $n_{b}=3$ [12],
therefore we model the base’s state as

${\bf q}_{b}=[x\quad y\quad h]^{T},$ (3)

where $x$ and $y$ are the linear displacements of the base’s center of mass
(COM) with respect to some fixed frame $\mathcal{F}_{f}$ (e.g. the starting
configuration), that are indeed equivalent to prismatic joints; $h$ is the
base’s angular displacement with respect to $\mathcal{F}_{f}$, i.e. the
heading, that is indeed equivalent to a revolute joint. The other degrees of
mobility (altitude, pitch and roll) are constrained by the geometry of the
terrain surface, that here, for the sake of simplicity, we assume to be flat.

In order to address the problem with DDP, $t$ is discretized into $N$ samples
and each element of $\boldsymbol{\nu}$ into $N_{k}$ samples, with
$k=0,\ldots,r$. Thus, the problem becomes an optimal path search one, on a
grid in $t_{i}$ and $\boldsymbol{\nu}_{\bf j}$, where ${\bf
j}\in\\{\left[1,N_{1}\right]\times\ldots\times\left[1,N_{r}\right]\\}\subset\mathbb{N}^{r}$
is the vector of redundancy parameter indices. In the resulting grid, the goal
is to find ${\bf q}(t_{i})={\bf q}(i)$, that optimizes the global cost

$I(N)=l({\bf q}(0))+\sum_{i=1}^{N}\phi\big{(}{\bf q}(i),\dot{\bf
q}(i)\big{)}.$ (4)

In (4), $l({\bf q}(0))$ is the cost associated to the initial configuration
and $\phi$ is the objective function to optimize along the trajectory, which,
for simplicity, is assumed to be only a function of joint positions ${\bf q}$
and velocities $\dot{\bf q}$. Therefore, the optimization problem, in a
recursive form, is

$\begin{split}I_{opt}(0)&=l({\bf q}(0)),\\\ I_{opt}(i)&=\min_{{\bf
q}(i-1)\in\mathcal{B}(i-1)}\Bigr{[}\phi\big{(}{\bf q}(i),\dot{\bf
q}(i))+I_{opt}(i-1)\Bigl{]},\end{split}$ (5)

where $\mathcal{B}(i)$, for each $i$, is the feasible domain of ${\bf q}(i)$,
and depends on the imposed constraints (see Section III); $I_{opt}$ is the
optimal return function, hence $I_{opt}(N)$ is the final optimal cost. It is
worth noting that, in (5), at each iteration, both $\dot{\bf q}(i)$, because
of discrete approximations of derivatives, and $I_{opt}(i-1)$ depend on the
choice made for ${\bf q}(i-1)$.

###  II-B Optimal interpolation

The joint space trajectory ${\bf q}_{opt}(t_{i})$ provided by DDP is
discretization-optimal, i.e. it is the global optimum in the discrete domain
defined by the indices $i$ and ${\bf j}$. To control the planning time, $N$
and $N_{k}$, i.e. the span of $i$ and ${\bf j}$, respectively, cannot be
arbitrarily large. In turn, this yields non-smooth trajectories, compromising
feasibility on real systems. To cope with this, we design a smoothing
algorithm based on optimal interpolation. To impose the task constraint, such
interpolation is performed in the domain of the spline control points
associated with the redundancy parameters $\boldsymbol{\nu}(i)$. Once this is
done, the other joint variables are retrieved through (2). Thus, said $t_{f}$
the task duration, the optimization problem is formalized as follows:

$\displaystyle\min_{\boldsymbol{\nu}(0),\ldots,\boldsymbol{\nu}({N})}$
$\displaystyle\int_{0}^{t_{f}}\phi(\tilde{\bf q},\dot{\tilde{\bf q}}),$ (6)
$\displaystyle s.t.\quad$ $\displaystyle{\bf q}(i)={\bf k}_{\alpha}^{-1}({\bf
x}(i),\boldsymbol{\nu}(i))\quad\forall i=1,\ldots,N$ (7)
$\displaystyle\tilde{\bf q}(t)=interp({\bf q})$ (8) $\displaystyle\tilde{\bf
q}(t)\in\mathcal{B}(t)\quad\forall t$ (9)

where the objective function $\phi$ is evaluated on the interpolated
trajectory $\tilde{{\bf q}}$; the decision variables $\boldsymbol{\nu}(i)$,
with $i=1,...,N$, are initialized with the result of DDP; inverse kinematics
(7) exploits the decision variables $\boldsymbol{\nu}(i)$ to find the
corresponding joint configurations at each $i$; the interpolation function
$interp$ in (8) embeds both linear interpolation for $x$, $y$ and $h$ and 3rd-
order B-spline interpolation for all the other joints; (9) verifies that joint
configurations respect first- and second-order derivatives embedded in the
sets $\mathcal{B}(t)$, discussed in Section III. With regards to the $interp$
function in (8), B-spline interpolation is preferred over simple polynomials
because it allows for an easier manipulation of the curves [24, 25]. However,
if applied to the base’s degrees of freedom, it might lead to overshoots that
would be difficult to control on a non-holonomic base. For this reason, in
this work, linear interpolation is preferred for the base variables, resulting
in turn-in-place maneuvers at non-differentiable points between linear
segments. Nevertheless, more sophisticated interpolation functions may be
adopted in (8) without changing the general scheme.

We note that the objective function $\phi$ in (6) is the same as DDP, as
defined in (4), and, likewise, it is subject to integral optimization. The
optimization problem (6)-(9) is non-convex, therefore most well-known
numerical optimization techniques can only guarantee local optimality.
Nevertheless, in our strategy, DDP “informs” this local search by providing an
optimal seed resulting from a global search, aiding both feasibility and
optimality.

##  III Constraints formulation

In our planning use case, we consider the following constraints to be
satisfied along the entire trajectory: end-effector path, i.e. forward
kinematics constraint; joint position, velocity and acceleration limits;
avoidance of obstacles and forbidden areas; pure rolling constraint, assuming
a non-holonomic base. Then, at the final waypoint, we also consider the
additional constraint that the target must not lie in the robot’s shade, so as
to guarantee the target illumination and improve its visibility from onboard
cameras.

###  III-A Joint limits and pure rolling constraint

Joint limits for the arm are formalized as

$\displaystyle\underline{{\bf q}_{a}}\leq$ $\displaystyle{\bf
q}_{a}\leq\overline{{\bf q}_{a}},$ (10) $\displaystyle\underline{\dot{{\bf
q}}_{a}}\leq$ $\displaystyle\dot{{\bf q}}_{a}\leq\overline{\dot{{\bf
q}}_{a}},$ (11) $\displaystyle\underline{\ddot{{\bf q}}_{a}}\leq$
$\displaystyle\ddot{{\bf q}}_{a}\leq\overline{\ddot{{\bf q}}_{a}},$ (12)

while, for the mobile base:

$\begin{split}\sqrt{\dot{x}^{2}+\dot{y}^{2}}\leq v_{max},\\\
|\dot{h}|\leq\omega_{max},\end{split}$ (13)

where $v_{max}$ and $\omega_{max}$ are the maximum tangential and angular
velocities of the base, respectively.

If a non-holonomic base is considered, the pure rolling constraint is

$\tan(h)=\frac{dy}{dx}.$ (14)

###  III-B Collision and forbidden areas avoidance

Let ${\bf q}_{p}(t)=\left[q_{1}(t),\ldots,q_{p}(t)\right]$ be the vector of
joint positions up to the $p$-th joint at time $t$ and $V_{p}\big{(}{\bf
q}_{p}(t)\big{)}$ be the convex hull of the $p$-th link at same time. Let
$V\big{(}{\bf q}(t)\big{)}$ (without a subscript) be the non-connected volume
occupied by the robot at time $t$, in the joint configuration ${\bf q}(t)$,
formalized as the union of the single links’ connected volumes, i.e.

$V\big{(}{\bf q}(t)\big{)}=\bigcup_{p=1}^{n}V_{p}\big{(}{\bf
q}_{p}(t)\big{)}.$ (15)

Let $V_{e}$ be the possibly non-connected volume of static environment
obstacles. Collision and self-collision constraints are respectively
formalized as

$\displaystyle V\big{(}{\bf q}(t)\big{)}\cap V_{e}=\emptyset\quad\forall t,$
(16) $\displaystyle V\big{(}{\bf q}_{p}(t)\big{)}\cap V\big{(}{\bf
q}_{k}(t)\big{)}=\emptyset\quad\forall t,p,k=1,\ldots,n\text{ with }p\neq k.$
(17)

Let $\mathcal{C}_{FA}$ be the set of $(x,y)$ coordinates corresponding to
forbidden areas in a 2-D occupancy map, usually acquired by the means of
visual systems. Then, the non-traversable areas avoidance constraint is
verified as:

$(x(t),y(t))\notin\mathcal{C}_{FA}\quad\forall t$ (18)

where $x(t)$ and $y(t)$, following their definition in (3), are the
coordinates of the mobile base at the time instant $t$.

###  III-C Overshadowing avoidance at the final waypoint

In Fig. 2, all the reference frames involved in the proposed use case are
reported. $\mathcal{F}_{w}$ is the _world_ reference frame, lying on the
terrain; $\mathcal{F}_{b}$ is the robot’s _base link_ reference frame;
$\mathcal{F}_{c}$ is the _camera_ reference frame, rigidly attached to the
mobile base; $\mathcal{F}_{s}$ is the _sun_ reference frame whose relative
position to $\mathcal{F}_{w}$ is given in terms of azimuth and elevation;
$\mathcal{F}_{t}$ is the frame fixed with the _target_.

Figure 2: Planning scenario with the indication of reference frames used for
planning; $\eta$ is the line passing through the origins of $\mathcal{F}_{c}$
and $\mathcal{F}_{t}$, $\gamma$ that passing through the origins of
$\mathcal{F}_{s}$ and $\mathcal{F}_{t}$.

The overshadowing avoidance constraint is verified by checking that the volume
of the robot $V({\bf q}(t_{f}))$ at the final waypoint does not intersect the
straight line $\gamma$ connecting $\mathcal{F}_{s}$ to $\mathcal{F}_{t}$:

$V({\bf q}(t_{f}))\cap\gamma=\emptyset.$ (19)

###  III-D Complete constraints formulation

Let us define the set $\mathcal{A}(t)$, containing all joints’ configurations
respecting the geometric (positional) constraints, i.e.:

$\mathcal{A}(t)=\Bigr{\\{}{\bf
q}(t):\eqref{eq:joint_pos_limits},\eqref{eq:env_collision_constraints},\eqref{eq:self_collision_constraints},\eqref{eq:forbidden_areas_constraint}\text{
hold}\Bigl{\\}}.$ (20)

At the final waypoint, $\mathcal{A}(t_{f})$ also accounts for (19), i.e.:

$\mathcal{A}(t_{f})=\Bigr{\\{}{\bf
q}(t_{f}):\eqref{eq:joint_pos_limits},\eqref{eq:env_collision_constraints},\eqref{eq:self_collision_constraints},\eqref{eq:forbidden_areas_constraint},\eqref{eq:shadow_constr}\text{
hold}\Bigl{\\}}.$ (21)

Then, the set $\mathcal{B}(t)$, employed in (5) and (9), contains the joint
configurations in $\mathcal{A}$ that also allow satisfying constraints on
first- and second-order derivatives, i.e. the reachable nodes at time $t$:

$\mathcal{B}(t)=\Bigr{\\{}{\bf
q}(t)\in\mathcal{A}(t):\eqref{eq:joint_vel_limits},\eqref{eq:joint_acc_limits},\eqref{eq:base_joint_limits},\eqref{eq:pure_rolling_constr}\text{
hold}\Bigl{\\}}.$ (22)

Here, to verify the differential constraints on positions, the derivatives in
(11), (12), (13), (14) are approximated with the Euler method, or equivalent
discrete approximation.

##  IV Objective Function formulation

As introduced in Section I, the objective function to optimize in our use case
is associated to the target visibility (TV) from a base-mounted camera.
Without loss of generality, and only for the sake of simplicity, we assume
that the camera is fixed during the entire trajectory, which is a typical
choice for navigation/localization cameras, thus a fixed transformation
$\mathcal{T}_{c}^{b}$ exists between the base and the camera frames, so that
the camera position ${\bf x}_{c}(t)$ in the world frame $\mathcal{F}_{w}$ can
be computed, at each time instant, from the transformation matrix

$\mathcal{T}_{c}^{w}(t)=\mathcal{T}_{b}^{w}\big{(}{\bf
q}_{b}(t)\big{)}\mathcal{T}_{c}^{b},$ (23)

where $\mathcal{T}_{b}^{w}\big{(}{\bf q}_{b}(t)\big{)}$ is the matrix
transforming $\mathcal{F}_{b}$ into $\mathcal{F}_{w}$. With reference to Fig.
2, the cost function is formalized as the dot product between the $x$-axis of
the camera frame (identified by the unit vector $\hat{\boldsymbol{x}}$) and
the unit vector $\hat{\boldsymbol{\eta}}$ lying onto the line $\eta$
connecting the camera to the target:

$\zeta_{TV}\big{(}{\bf
q}\big{)}=1-\hat{\boldsymbol{x}}\cdot\hat{\boldsymbol{\eta}}.$ (24)

Note that, since we are solving a minimization problem, the 1-complement of
the dot product is considered. In order to also penalize solutions requiring
high energy consumption, which is crucial for space robots, the squared norm
of joint velocities $\zeta_{SNV}$ is considered:

$\zeta_{SNV}=\dot{{\bf q}}^{T}\dot{{\bf q}}.$ (25)

This index limits unnecessary joint movements and is often used as a simple
surrogate of energy consumption when an accurate energy model is not available
[26]. Therefore, the objective function $\phi$ in (5) and (6) is multi-
objective and is formally defined as:

$\phi=\sigma\zeta_{TV}+(1-\sigma)\zeta_{SNV},$ (26)

where $0\leq\sigma\leq 1$ weighs the two performance indices.

##  V 3DROCS

In order to demonstrate that OptiWB is suitable to be employed in real mission
planning scenarios, in this section, we discuss its integration with 3DROCS, a
Rover Operations Control Software by Trasys International, designed to
prepare, monitor and control rover planetary surface operations, with the help
of a 3D graphical environment. It includes models of robotic systems and the
environment in which they operate, e.g. terrain models, textures, lighting and
on-line shadowing, relief maps to allow the scientific and engineering ground
operators to indicate and verify sites of interest and dangerous areas that
are beyond the limits of the rover’s capabilities. It also provides mission-
specific functions, such as Activity Plans (AP) specification, predictions
related to power, communication and mechanical subsystems.

As far as planning is concerned, 3DROCS allows setting targets that are
locations of scientific interest on the terrain, defining paths to be followed
by the rover, and indicating forbidden areas. However, the optimality problem
of the rover’s motion is not addressed directly for the required paths, but
the planning system relies on a simple decoupled approach where the mobile
base path is planned first, followed by motion planning of the arm, when the
target becomes proximal. This approach cannot guarantee optimal movements and
might require complex maneuvers in the proximity of the target to satisfy some
application constraints, like optimal target visibility and shadowing. In this
context, we propose to integrate OptiWB with 3DROCS to fill this gap.

Since the concerned processes exist in the realm of offline ground planning,
the integration only requires the definition of suitable data formats and
modules to read and write interface data files, e.g. json. In particular, the
environment description $V_{e}$, i.e. obstacle specifications, the forbidden
areas $\mathcal{C}_{FA}$, sun position in terms of azimuth $\psi$ and
elevation $\theta$, the target position ${\bf x}_{t}$, and the task space
trajectory ${\bf x}(t)$ are specified by 3DROCS. Then, OptiWB returns the
optimal joint space trajectory $\tilde{\bf q}_{opt}(t)$. These interfaces are
summarized in Fig. 3.

Figure 3: Data interface between 3DROCS and OptiWB.

##  VI Experimental results

Experiments are performed on a simplified version of the Analog-1 Interact
rover [21] composed of a four-wheel moving platform and a LBR iiwa 14 R820
7-DOF arm [22] mounted on the front panel, and slightly shifted to the right
with respect to the center of the panel. This configuration provides 10 DOFs,
as altitude, roll and pitch of the base are assumed to be constrained by the
contacts with the flat ground. For the sake of simplicity, a task with $m=7$
is assigned, yielding $r=3$. The task constrains the end-effector position, as
well as one of the arm’s joints, i.e. the third joint. The redundancy
parameters are selected to be the vector ${\bf q}_{b}$, as in [12]. The
assigned task space trajectory has 40 waypoints, a duration of $t_{f}=39$ s
and extends for $3$ m along the $x$-axis, $1.5$ m along the $y$-axis and $0.9$
m along the $z$-axis, as shown in Fig. 4. The planning scenario is the one of
Fig. 2.

Figure 4: Assigned task space trajectory (solid, blue) and its projection on
the $x$-$y$ plane (dashed, black). Frames for each waypoint represent the end-
effector orientation.

The DDP planner is implemented in ROS, while the optimal interpolation
procedure is implemented in MATLAB, solving (6) with the interior point
method. The task space trajectory is inverted with the DDP algorithm using a
grid resolution of $0.1$ m for $x$ and $y$, and $0.2$ rad for $h$. In (26),
$\sigma=0.95$. The planned joint space trajectory and the corresponding base
path in the $x$-$y$ plane are provided in Fig. 5.

(a)

(b)

Figure 5:  (a) Robot joint trajectory ${\bf q}(t)$ planned with DDP and (b)
corresponding base path in the $x$-$y$ plane. Red arrows represent the base
heading while circles represent the points where the base stops and performs a
turn-in-place maneuver.

The cost of the final solution is $I_{opt}(N)=1.5829$, with integral cost for
each performance index equal to $I_{TV}=0.9695$ and $I_{SNV}=13.2399$. We can
notice that, due to DDP discretization, the joint positions are not smooth.
Also, the heading $h$ exhibits rapid changes and the base frequently stops at
discrete configurations (see circles in Fig. 5(b)). This trajectory is not
suitable to generate controller references onboard, as feasibility would be
likely to be compromised. Therefore, the solution ${\bf q}_{opt}(t_{i})$ of
DDP is used as the initial guess, i.e. initial spline control points, for
optimal interpolation in (6)-(9), generating smooth (and optimal) arm and base
trajectories $\tilde{\bf q}_{opt}(t)$. They are reported in Fig. 6(a) and Fig.
6(b), respectively.

(a)

(b)

Figure 6:  (a) Robot joint trajectory $\tilde{\bf q}_{opt}(t)$ found after
optimal interpolation and (b) corresponding base path in the $x$-$y$ plane.

The final cost of the post-processed solution is $I_{opt}(N)=1.002$, with
$I_{TV}=0.8109$ and $I_{SNV}=4.6332$. Comparing this cost with the
intermediate one from DDP, we note that, although B-spline interpolation sets
higher-order constraints (not included in the problem formulation with DDP),
the cost for both performance indices decreases. In addition, as desired, the
solution is smooth, which is a crucial aspect for trajectory feasibility at
control level.

The accompanying video [27] shows the DDP and interpolated trajectories in
RViz. At the last waypoint, the robot never stands between the sun and the
target, and the base motion is planned in a way that the target is kept in the
center of the camera’s visual cone as much as possible during trajectory
execution (Fig. 7). On the other hand, for the same use case, if no
constraints are taken into account during planning, forbidden areas violation
and target shadowing are possible. This is shown in the second section of the
video: at the beginning of the trajectory, the forbidden area is traversed
while, at the end, the base overshadows the target.

Once the optimal plan is available, it is returned to 3DROCS, providing a more
detailed description of the environment, for rehearsal and validation. The
last section of the video shows the plan execution in the 3DROCS simulator: as
required, the target always lies in the camera vision cone (in blue), the
robot’s shadow does not cover the target and the red forbidden areas are
avoided.

Figure 7: View of the target on the $u$-$v$ image plane when executing the
interpolated trajectory. The axes are in meters and correspond to the CCD
size.

##  VII Conclusions

Robot motion planning for space exploration is of fundamental importance
during mission planning and operations. Most state-of-the-art tools used at
ground control centers do not provide planning modules effectively exploiting
kinematic redundancy to optimize performance indices of interest while
satisfying constraints. We proposed an Optimal Whole Body Planner (OptiWB)
performing a two-step optimization based on Discrete Dynamic Programming (DDP)
and optimal interpolation. Focusing on a planetary mobile manipulator use
case, constraints typical of space exploration are formalized, such as
collision, forbidden areas, and target overshadowing avoidance. In addition, a
performance index based on the target visibility from navigation cameras is
considered. The effectiveness and applicability of OptiWB are demonstrated by
integrating it with a commercial mission planner, 3DROCS, employed in several
ESA missions. The results show that a feasible and optimal plan is generated
satisfying the application constraints. At the same time, the ground operator
is relieved from designing cumbersome rover and arm maneuvers: the whole-body
approach simplifies the planning process, as non-trivial movements can be
planned from the sole specification of a desired task-space trajectory.

We envisage to extend our planning system towards more challenging scenarios
of planetary construction, where e.g., the system is characterized by many
degrees of redundancy. On the technological side, we aim at integrating OptiWB
within the new born Space ROS ecosystem.

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# On the degree of approximation of continuous functions by a linear
transformation of their Fourier series

Xhevat Z. Krasniqi  Faculty of Education, University of Prishtina “Hasan
Prishtina”, Avenue “Mother Theresa” 5, 10000 Prishtina, Republic of Kosovo
<EMAIL_ADDRESS>
###### Abstract

In this paper, we have proved four theorems on the degree of approximation of
continuous functions by matrix means of their Fourier series which is
expressed in terms of the modulus of continuity and a non-negative mediate
function.

###### keywords:

Degree of approximation, Fourier series, modulus of continuity, mediate
function.

42A24, 41A25. 30 https://doi.org/10.46298/cm.9273

##  1 Introduction

By $S_{n}(f;x)$ we denote the $n$-th partial sum of the Fourier series of a
$2\pi$-periodic continuous function $f(x)$ at $x$ and
$\omega(\delta)=\omega(\delta,f)$ the modulus of continuity of $f$. In short,
we denote by $A:=(a_{n,k})$, $(k,n=0,1,\dots)$ a lower triangular infinite
matrix of real numbers and the $A$-transform of $\\{S_{n}(f;x)\\}$ by

$T_{n,A}(f;x):=\sum_{k=0}^{n}a_{n,k}S_{k}(f;x),\quad(n=0,1,\dots).$

Throughout this paper we write $u=\mathcal{O}(v)$ if there exists a positive
constant $C$ such that $u\leq Cv$, and $\|\cdot\|$ denotes the sup-norm. P.
Chandra [CH1], [CH2] was the first who estimated the deviation $T_{n,A}(f)-f$,
in sup-norm, when $\\{a_{n,k}\\}$ is a monotonic sequences with respect to
$k$. Many years later, L. Leindler [L] replaced the monotonicity condition
with the so-called _rest bounded variation sequences_ and _“head bounded
variation sequences”_ (these will be recalled later in this paper). Chandra’s
theorems are the following.

######  Theorem 1.1.

Let $\\{a_{n,k}\\}$ satisfy the following conditions:

$a_{n,k}\geq 0\qquad\mbox{and}\qquad\sum_{k=0}^{n}a_{n,k}=1,$ (1) $a_{n,k}\leq
a_{n,k+1}\quad(k=0,1,\dots,n-1;n=0,1,\dots).$ (2)

Suppose $\omega(t)$ is such that

$\int_{u}^{\pi}t^{-2}\omega(t)dt=\mathcal{O}\left(H(u)\right)\quad(u\to+0),$
(3)

where $H(u)\geq 0$ and

$\int_{0}^{t}H(u)du=\mathcal{O}\left(tH(t)\right)\quad(t\to+0).$ (4)

Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,n}H(a_{n,n})\right).$

######  Theorem 1.2.

Let (1), (2) and (3) hold. Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\omega(\pi/n)\right)+\mathcal{O}\left(a_{n,n}H(\pi/n)\right).$

If, in addition, $\omega(t)$ satisfies (4) then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,n}H(\pi/n)\right).$

######  Theorem 1.3.

Let us assume that (1) and

$a_{n,k}\geq a_{n,k+1}\quad(k=0,1,\dots,n-1;n=0,1,\dots)$ (5)

hold. Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\omega(\pi/n)+\sum_{k=1}^{n}k^{-1}\omega(\pi/k)\sum_{r=0}^{k+1}a_{n,r})\right).$

######  Theorem 1.4.

Let (1), (3), (4) and (5) hold. Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,0}H(a_{n,0})\right).$

To obtain his results, L. Leindler [L] used the following classes of
sequences. A sequence $\textbf{c}:=\\{c_{n}\\}$ of nonnegative numbers tending
to zero is called of Rest Bounded Variation, or briefly $\textbf{c}\in RBVS$,
if it has the property

$\sum_{n=m}^{\infty}|c_{n}-c_{n+1}|\leq K(\textbf{c})c_{m}$

for all natural numbers $m$, where $K(\textbf{c})$ is a constant depending
only on c. A sequence $\textbf{c}:=\\{c_{n}\\}$ of nonnegative numbers will be
called of Head Bounded Variation, or briefly $\textbf{c}\in HBVS$, if it has
the property

$\sum_{n=0}^{m-1}|c_{n}-c_{n+1}|\leq K(\textbf{c})c_{m}$

for all natural numbers $m$, or only for all $m\leq N$ if the sequence c has
only finite nonzero terms, and the last nonzero term is $c_{N}$. To go further
it is clear that condition $0<K(\textbf{c})\leq K$ needs to be assumed, where
$K$ is a positive constant. Assuming that for all $n$ and $0\leq m\leq n$

$\sum_{k=m}^{\infty}|a_{n,k}-a_{n,k+1}|\leq Ka_{n,m}$ (6)

and

$\sum_{k=0}^{m-1}|a_{n,k}-a_{n,k+1}|\leq Ka_{n,m}$ (7)

hold, where $K$ is a positive constant, Leindler [L] proved the following:

######  Theorem 1.5.

The statements of Theorems 1.1, 1.2, 1.3 and 1.4 hold with (7) in place of
(2), and with (6) in place of (5), respectively; naturally maintaining all the
other assumptions.

These results are extended further in [XhK], then they are generalized in
[XhK1], and are treated again in [XhK2] using some other means (see also
[BD]). Very recently, W. Lenski and B. Szal [LS], used (for trigonometric
approximation in the variable space $L^{p(x)}$ and $m=n$) the following
condition

$\sum_{k=0}^{m-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),\qquad(\beta\geq
0),$ (8)

assumed on the sequence $\\{a_{n,k}\\}$, $0\leq m\leq n$. To my best
knowledge, such a condition appears for the first time in the mathematical
literature, which for $\beta=0$ coincides with condition (7). Motivated by the
above condition (for $a_{n,m}=\frac{1}{n+1}$ and $m=n$ this condition has been
introduced in [LS]) we introduce another new condition

$\sum_{k=m}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),\qquad(\beta\geq
0),$ (9)

on the sequence $\\{a_{n,k}\\}$, $0\leq m\leq n$, which for $\beta=0$ coincide
with condition (6).

######  Remark 1.6.

Note that for $\beta=0$ these two conditions coincides with (7) and (6),
respectively. Also, it was shown in [LS] that if
$\\{{(k+1)}^{-\beta}a_{n,k}\\}\in HBVS$, then condition $\eqref{star}$ holds
true. Moreover, it easy to verify that condition $\eqref{starstar}$ implies
$\\{{(k+1)}^{-\beta}a_{n,k}\\}\in RBVS$. Furthermore, some examples, which
supports these conditions, are given there as well.

Now we are concerned weather these two conditions on the sequence
$\\{a_{n,k}\\}$ can be used in Theorems 1.1–1.4, instead of (2) and (1.1),
which is in fact the aim of this paper.

##  2 Auxiliary Results

The next lemmas play a helpful role for the proof of our results.

######  Lemma 2.1.

[CH2] If (3) and (4) hold then

$\int_{0}^{v}t^{-1}\omega(t)dt=\mathcal{O}\left(vH(v)\right)\quad(v\to+0).$

######  Lemma 2.2.

If (3) and (4) hold then

$\int_{0}^{\pi/m}\omega(t)dt=\mathcal{O}\left(m^{-2}H(\pi/m)\right).$

######  Proof 2.3.

Using Lemma 2.1 we have

$\displaystyle\int_{0}^{\pi/m}\omega(t)dt$
$\displaystyle=\int_{0}^{\pi/m}\frac{t\omega(t)}{t}dt\leq\frac{\pi}{m}\int_{0}^{\pi/m}\frac{\omega(t)}{t}dt$
$\displaystyle=\frac{\pi}{m}\mathcal{O}\left(\frac{\pi}{m}H\left(\frac{\pi}{m}\right)\right)=\mathcal{O}\left(m^{-2}H(\pi/m)\right),$

which completes the proof.

######  Lemma 2.4.

[LS] If $\beta\geq 0$ and $0<t\leq\pi$, then

$\left|\sum_{j=0}^{m}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|\leq\frac{\pi^{2}{(m+1)}^{\beta}}{t^{2}}.$

######  Lemma 2.5.

Let $\beta\geq 0$ be a real number. If for a fixed $n$, the sequence
$\\{a_{n,k}\\}$ satisfies the condition

$\sum_{k=m}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for $0\leq m\leq n$, then uniformly in $0<t\leq\pi$,

$|K_{n}(t)|:=\left|\sum_{k=0}^{n}a_{n,k}\frac{\sin\left(k+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|=\mathcal{O}\left(t^{-1}A_{n,\tau}\right),$

where $\displaystyle A_{n,\tau}:=\sum_{r=0}^{\tau}a_{n,r}$ and $\tau$ denotes
the integer part of $\,\frac{\pi}{t}$. If $\\{a_{n,k}\\}$ satisfies the
condition

$\sum_{k=0}^{m-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for $0\leq m\leq n$, then

$|K_{n}(t)|=\mathcal{O}\left(\frac{a_{n,n}}{t^{2}}\right).$

######  Proof 2.6.

Assume that $n\geq\tau$. Applying summation by parts, the inequalities
$|\sin\gamma|\leq 1$ and $\pi\sin\gamma\geq 2\gamma$, ($\gamma\in[0,\pi/2]$),
we have

$\displaystyle\left|K_{n}(t)\right|$
$\displaystyle\leq\sum_{k=0}^{\tau}\frac{a_{n,k}}{2\sin\frac{t}{2}}+\left|\sum_{k=\tau+1}^{n}a_{n,k}\frac{\sin\left(k+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\leq\frac{\pi}{2t}\sum_{k=0}^{\tau}a_{n,k}+\left|\sum_{k=\tau+1}^{n}\frac{a_{n,k}}{{(k+1)}^{\beta}}{(k+1)}^{\beta}\frac{\sin\left(k+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\leq\frac{a_{n,\tau+1}}{{(\tau+2)}^{\beta}}\left|\sum_{j=0}^{\tau}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\quad+\frac{\pi}{2t}\sum_{k=0}^{\tau}a_{n,k}+\frac{a_{n,n}}{{(n+1)}^{\beta}}\left|\sum_{j=0}^{n}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\quad+\left|\sum_{k=\tau+1}^{n-1}\left(\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right)\sum_{j=0}^{k}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|.$

Now, using Lemma 2.4 and our assumption, we get

$\displaystyle\left|K_{n}(t)\right|$
$\displaystyle\leq\frac{a_{n,\tau+1}}{{(\tau+2)}^{\beta}}\frac{\pi^{2}{(\tau+1)}^{\beta}}{t^{2}}+\frac{\pi}{2t}\sum_{k=0}^{\tau}a_{n,k}$
$\displaystyle\quad+\frac{a_{n,n}}{{(n+1)}^{\beta}}\frac{\pi^{2}{(n+1)}^{\beta}}{t^{2}}+\sum_{k=n}^{\infty}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|\frac{\pi^{2}{(k+1)}^{\beta}}{t^{2}}$
$\displaystyle=\frac{\pi}{2t}\sum_{k=0}^{\tau}a_{n,k}+\frac{\pi^{2}}{t^{2}}\mathcal{O}(a_{n,\tau}+a_{n,n}).$

But (by assumption) we also get

$\displaystyle a_{n,n}$
$\displaystyle=\sum_{k=n}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|$
$\displaystyle\leq\sum_{k=\tau}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,\tau}).$

Therefore,

$\left|K_{n}(t)\right|=\frac{\pi}{2t}\sum_{k=0}^{\tau}a_{n,k}+\frac{\pi}{t}\mathcal{O}((\tau+1)a_{n,\tau})=\mathcal{O}\left(t^{-1}\sum_{k=0}^{\tau}a_{n,k}\right),$

since for $0\leq k\leq\tau$ we have

$\displaystyle a_{n,\tau}$
$\displaystyle\leq\sum_{i=\tau}^{\infty}{(i+1)}^{\beta}\left|\frac{a_{n,i}}{{(i+1)}^{\beta}}-\frac{a_{n,i+1}}{{(i+2)}^{\beta}}\right|$
$\displaystyle\leq\sum_{i=k}^{\infty}{(i+1)}^{\beta}\left|\frac{a_{n,i}}{{(i+1)}^{\beta}}-\frac{a_{n,i+1}}{{(i+2)}^{\beta}}\right|=\mathcal{O}(a_{n,k}).$

Similarly, we have obtained

$\displaystyle\left|K_{n}(t)\right|$
$\displaystyle\leq\frac{a_{n,n}}{{(n+1)}^{\beta}}\left|\sum_{j=0}^{n}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\quad+\left|\sum_{k=0}^{n-1}\left(\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right)\sum_{j=0}^{k}{(j+1)}^{\beta}\frac{\sin\left(j+\frac{1}{2}\right)t}{2\sin\frac{t}{2}}\right|$
$\displaystyle\leq\frac{a_{n,n}}{{(n+1)}^{\beta}}\frac{\pi^{2}{(n+1)}^{\beta}}{t^{2}}+\sum_{k=0}^{n-1}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|\frac{\pi^{2}{(k+1)}^{\beta}}{t^{2}}$
$\displaystyle=\mathcal{O}\left(\frac{a_{n,n}}{t^{2}}\right).$

The proof is now complete.

##  3 Main Results

We start with the following.

######  Theorem 3.1.

Let $\\{a_{n,k}\\}$ satisfy the conditions (1) and

$\sum_{k=0}^{m-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for all $m=0,1,\dots,n;\,(n=0,1,\dots)$. Suppose $\omega(t)$ is such that (3)
and (4) hold, then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,n}H\left(a_{n,n}\right)\right).$ (10)

######  Proof 3.2.

Writing

$\psi_{x}(t):=\frac{1}{2}\left\\{f(x+t)+f(x-t)-2f(x)\right\\},$

we get

$T_{n,A}(f;x)-f(x)=\frac{2}{\pi}\int_{0}^{\pi}\psi_{x}(t)K_{n}(t)\,dt$

and

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\int_{0}^{a_{n,n}}|\psi_{x}(t)||K_{n}(t)|\,dt+\int_{a_{n,n}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt\right).$
(11)

By the well-known inequality $\sin\theta\geq\frac{2}{\pi}\theta$ for
$0\leq\theta\leq\frac{\pi}{2}$, $|\sin\theta|\leq 1$, and Lemma 2.1 we get

$\displaystyle\int_{0}^{a_{n,n}}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}(1)\int_{0}^{a_{n,n}}t^{-1}\omega(t)dt$
$\displaystyle=\mathcal{O}\left(a_{n,n}H\left(a_{n,n}\right)\right).$ (12)

Now using Lemma 2.5 and (3), we have

$\displaystyle\int_{a_{n,n}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,n}\right)\int_{a_{n,n}}^{\pi}t^{-2}\omega(t)dt$
$\displaystyle=\mathcal{O}\left(a_{n,n}H\left(a_{n,n}\right)\right).$ (13)

Substituting (3.2) and (3.2) into (11), this implies (10). The proof is now
complete.

######  Theorem 3.3.

Let $\\{a_{n,k}\\}$ satisfy the conditions (1) and

$\sum_{k=0}^{m-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for all $m=0,1,\dots,n;\,(n=0,1,\dots),$ and (3). Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\omega\left(\frac{\pi}{n+1}\right)+a_{n,n}H\left(\frac{\pi}{n+1}\right)\right).$
(14)

If, in addition, $\omega(t)$ satisfies (4) then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,n}H\left(\frac{\pi}{n+1}\right)\right).$
(15)

######  Proof 3.4.

Reasoning as in the proof of Theorem 3.1 we have

$\displaystyle\|T_{n,A}-f\|$
$\displaystyle=\mathcal{O}\left(\frac{2}{\pi}\right)\int_{0}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}\left(\frac{2}{\pi}\right)\left(\\!\int_{0}^{\frac{\pi}{n+1}}|\psi_{x}(t)||K_{n}(t)|\,dt+\\!\int_{\frac{\pi}{n+1}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt\right).$
(16)

Using the inequalities $\sin\theta\geq\frac{2}{\pi}\theta$ for
$0\leq\theta\leq\frac{\pi}{2}$, $\displaystyle|\sin t|\leq t$ and (1) we have

$\displaystyle\int_{0}^{\frac{\pi}{n+1}}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}(n+1)\int_{0}^{\frac{\pi}{n+1}}\omega(t)\,dt$
$\displaystyle=\mathcal{O}\left(\omega\left(\frac{\pi}{n+1}\right)\right).$
(17)

Lemma 2.5 and (3) implies

$\displaystyle\int_{\frac{\pi}{n+1}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,n}\right)\int_{\frac{\pi}{n+1}}^{\pi}t^{-2}\omega(t)\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,n}H\left(\frac{\pi}{n+1}\right)\right).$
(18)

Putting (3.4) and (3.4) into (3.4) we obtain (14). Now we prove (15). It is
clear that for $0\leq l\leq n-1$ and by our assumption we have

$\displaystyle\mathcal{O}(a_{n,n})$
$\displaystyle=\sum_{k=0}^{n-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|$
$\displaystyle\geq{(l+1)}^{\beta}\sum_{k=l}^{n-1}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|$
$\displaystyle\geq{(l+1)}^{\beta}\left|\frac{a_{n,l}}{{(l+1)}^{\beta}}-\frac{a_{n,n}}{{(n+1)}^{\beta}}\right|$
$\displaystyle\geq{(l+1)}^{\beta}\left|\frac{a_{n,l}}{{(l+1)}^{\beta}}-\frac{a_{n,n}}{{(l+1)}^{\beta}}\right|\geq
a_{n,l}-a_{n,n},$

from which we conclude that

$a_{n,l}=\mathcal{O}\left(a_{n,n}\right).$

Using the last relation we get

$\displaystyle 1=\sum_{l=0}^{n}a_{n,l}$
$\displaystyle=\mathcal{O}\left(1\right)\sum_{l=0}^{n}a_{n,n}=\mathcal{O}\left((n+1)a_{n,n}\right)$
$\displaystyle\Longrightarrow\frac{1}{n+1}=\mathcal{O}\left(a_{n,n}\right).$
(19)

By Lemma 2.2, (3.4) and (3.4) we obtain

$\displaystyle\int_{0}^{\frac{\pi}{n+1}}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}(n+1)\int_{0}^{\frac{\pi}{n+1}}\omega(t)\,dt$
$\displaystyle=\mathcal{O}\left(n+1\right){(n+1)}^{-2}H\left(\frac{\pi}{n+1}\right)$
$\displaystyle=\mathcal{O}\left({(n+1)}^{-1}H\left(\frac{\pi}{n+1}\right)\right)$
$\displaystyle=\mathcal{O}\left(a_{n,n}H\left(\frac{\pi}{n+1}\right)\right).$
(20)

Thus (3.4) and (3.4) imply (15), which completes the proof.

######  Theorem 3.5.

Let us assume that (1) and

$\sum_{k=m}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for all $m=0,1,\dots,n;\,(n=0,1,\dots)$. Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\omega\left(\frac{\pi}{n+1}\right)+\sum_{v=1}^{n}v^{-1}\omega\left(\frac{\pi}{v}\right)\sum_{j=0}^{v}a_{n,j}\right).$

######  Proof 3.6.

Once again, we start from (3.4). For the first integral we use (3.4):

$\int_{0}^{\frac{\pi}{n+1}}|\psi_{x}(t)||K_{n}(t)|\,dt=\mathcal{O}\left(\omega\left(\frac{\pi}{n+1}\right)\right).$
(21)

Applying Lemma 2.5 we obtain

$\displaystyle\int_{\frac{\pi}{n+1}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}\left(\sum_{v=1}^{n}\int_{\frac{\pi}{v+1}}^{\frac{\pi}{v}}t^{-1}\omega(t)A_{n,\tau}dt\right)$
$\displaystyle=\mathcal{O}\left(\sum_{v=1}^{n}(v+1)\omega\left(\frac{\pi}{v}\right)\sum_{j=0}^{v+1}a_{n,j}\int_{\frac{\pi}{v+1}}^{\frac{\pi}{v}}dt\right)$
$\displaystyle=\mathcal{O}\left(\sum_{v=1}^{n}v^{-1}\omega\left(\frac{\pi}{v}\right)\sum_{j=0}^{v}a_{n,j}\right),$

since

$\int_{\frac{\pi}{v+1}}^{\frac{\pi}{v}}dt=\frac{\pi}{v(v+1)}.$

This with (21) and (3.4) completes the proof.

######  Theorem 3.7.

Let $\\{a_{n,k}\\}$ satisfy the conditions (1) and

$\sum_{k=m}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,m}),$

for all $m=0,1,\dots,n;\,(n=0,1,\dots),$ and suppose that $\omega(t)$ is such
that (3) and (4) hold. Then

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(a_{n,0}H\left(a_{n,0}\right)\right).$ (22)

######  Proof 3.8.

We have seen in the proof of Lemma 2.5 that

$\left|K_{n}(t)\right|\leq\frac{\pi^{2}a_{n,n}}{t^{2}}+\frac{\pi^{2}}{t^{2}}\sum_{k=0}^{n-1}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|.$

Based on our assumption, for $n\geq 0$, we have

$\displaystyle a_{n,n}$
$\displaystyle=\sum_{k=n}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|$
$\displaystyle\leq\sum_{k=0}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}(a_{n,0}),$

and thus

$\left|K_{n}(t)\right|\leq\frac{\pi^{2}a_{n,0}}{t^{2}}+\frac{\pi^{2}}{t^{2}}\sum_{k=0}^{\infty}{(k+1)}^{\beta}\left|\frac{a_{n,k}}{{(k+1)}^{\beta}}-\frac{a_{n,k+1}}{{(k+2)}^{\beta}}\right|=\mathcal{O}\left(\frac{a_{n,0}}{t^{2}}\right).$

Now we write

$\|T_{n,A}(f)-f\|=\mathcal{O}\left(\int_{0}^{a_{n,0}}|\psi_{x}(t)||K_{n}(t)|\,dt+\int_{a_{n,0}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt\right).$
(23)

Similarly, as in (3.2)), we have

$\displaystyle\int_{0}^{a_{n,0}}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}(1)\int_{0}^{a_{n,0}}t^{-1}\omega(t)\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,0}H\left(a_{n,0}\right)\right).$ (24)

Moreover, as in (3.2), we obtain

$\displaystyle\int_{a_{n,0}}^{\pi}|\psi_{x}(t)||K_{n}(t)|\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,0}\right)\int_{a_{n,0}}^{\pi}t^{-2}\omega(t)\,dt$
$\displaystyle=\mathcal{O}\left(a_{n,0}H\left(a_{n,0}\right)\right).$ (25)

Putting (3.8) and (3.8) into (23), we get (22), which completes the proof.

######  Remark 3.9.

Based on the remark given at the end of the first section (of this paper), we
conclude that our results hold true if we assume that either the conditions
$\\{{(k+1)}^{-\beta}a_{n,k}\\}\in HBVS$ or $\\{{(k+1)}^{-\beta}a_{n,k}\\}\in
RBVS$, in the appropriate cases.

######  Remark 3.10.

If we put $\beta=0$ in our theorems, then we exactly obtain all results proved
in [L].

##  4 Corollaries

Assume that $A:=(a_{n,k})$ is a matrix defined by

$a_{n,k}=\begin{cases}\frac{p_{n-k}}{P_{n}},&0\leq k\leq n\\\
0,&\text{otherwise},\end{cases}$

where $\\{p_{k}\\}_{k=0,1,\dots,n}$, is a sequence of non-negative numbers
with $P_{n}:=\sum_{k=0}^{n}p_{k}\neq 0$. In this case, the transform
$T_{n,A}(f;x)$ reduces to polynomials of the form

$T_{n,A}(f;x)\equiv
N_{n}(f;x)=\frac{1}{P_{n}}\sum_{k=0}^{n}p_{n-k}s_{k}(f;x).$

Consequently, we have the following.

######  Corollary 4.1.

Let all conditions of Theorem 3.7 be satisfied. Then

$\|N_{n}(f)-f\|=\mathcal{O}\left(\frac{p_{n}}{P_{n}}H\left(\frac{p_{n}}{P_{n}}\right)\right).$

If the matrix $A:=(a_{n,k})$ is defined as follows:

$a_{n,k}=\begin{cases}\frac{p_{k}}{P_{n}},&0\leq k\leq n\\\
0,&\text{otherwise},\end{cases}$

where $\\{p_{k}\\}_{k=0,1,\dots,n}$, is a sequence of non-negative numbers
with $P_{n}:=\sum_{k=0}^{n}p_{k}\neq 0$. Now, the transform $T_{n,A}(f;x)$ are
the polynomials

$T_{n,A}(f;x)\equiv R_{n}(f;x)=\frac{1}{P_{n}}\sum_{k=0}^{n}p_{k}s_{k}(f;x).$

Therefore, we have:

######  Corollary 4.2.

Let all conditions of Theorem 3.1 be satisfied. Then

$\|R_{n}(f)-f\|=\mathcal{O}\left(\frac{p_{n}}{P_{n}}H\left(\frac{p_{n}}{P_{n}}\right)\right).$

######  Remark 4.3.

For $\beta=0$, Corollary 4.1 and Corollary 4.2 reduce to two corollaries given
in [L] (see pages 108-109).

If $\omega(u)=u^{\alpha}$, $0<\alpha\leq 1$, and
$f\in\text{Lip}(\omega)\equiv\text{Lip}(\alpha)$, then
$\omega(t)=\mathcal{O}\left(\delta^{\alpha}\right),\,\delta\geq 0$. Hence, if
$f\in\text{Lip}(\alpha)$ and

$H(u)=\begin{cases}u^{\alpha-1},&0<\alpha<1;\\\
\log\left(\frac{\pi}{u}\right),&\alpha=1,\end{cases}$

and then from Corollary 4.2 we get:

######  Corollary 4.4.

Let $f\in C[0,2\pi]$, $f\in\text{Lip}(\alpha)$, $(0<\alpha\leq 1)$, and let
$\\{p_{k}/P_{n}\\}$ satisfies the condition $\eqref{star}$. Then the degree of
approximation of $f$ by $R_{n}(f;x)$ of its Fourier series is given by

$\|R_{n}(f)-f\|=\begin{cases}\mathcal{O}\left({\left(\frac{p_{n}}{P_{n}}\right)}^{\alpha}\right),&0<\alpha<1;\\\
\mathcal{O}\left(\frac{p_{n}}{P_{n}}\log\left(\frac{\pi
P_{n}}{p_{n}}\right)\right),&\alpha=1.\end{cases}$

######  Remark 4.5.

Note that for $\beta=0$ in Corollary 4.4 we obtain one of the theorems proved
in [CH4].

### Acknowledgements

The author expresses his great gratitude to the anonymous referee who with
her/his report draw my attention that Lemma 2.2 is a consequence of Lemma 2.1
offering a simple proof, and also made several comments which definitely
improved this paper. Also, I would like to thank the editor of Communications
in Mathematics who helped me with many advises pertaining to the language used
throughout the paper.

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13 April 202010 July 2020Karl Dilcher