Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

This paper considers actuator redundancy management for a class of overactuated nonlinear systems. Two tools for distributing the control effort among a redundant set of actuators are optimal control design and control allocation. In this paper, we investigate the relationship between these two design tools when the performance indexes are quadratic in the control input. We show that for a particular class of nonlinear systems, they give exactly the same design freedom in distributing the control effort among the actuators. Linear quadratic optimal control is contained as a special case. A benefit of using a separate control allocator is that actuator constraints can be considered, which is illustrated with a flight control example.

Resolving actuator redundancy: optimal control vs. control allocation

Linear motion commands of computer numerical control (CNC) machine tools need to be smoothed at the transition corners in order to guarantee continuous and steady machining. However, because of the complex kinematic constraints, very few researches have devoted to developing analytical and high order continuous corner smoothing algorithms of five-axis tool paths, although it is important to guarantee both high calculation efficiency and good dynamic performance of five-axis CNC machining. This paper develops an analytical C 3 continuous corner smoothing algorithm of five-axis tool paths by locally inserting specially designed quintic micro splines into the transition corners of five-axis linear commands. C 3 continuity of the tool tip position and the tool orientation are guaranteed along the entire tool path. The maximal approximation errors of the tool tip position and the tool orientation are both constrained in the workpiece coordinate system. The synchronization of the tool tip position and tool orientation are mathematically guaranteed at the junctions of the linear and spline segments. The proposed corner smoothing algorithm can calculate all control points of the locally inserted tool tip position and tool orientation splines analytically without any iteration, which makes it very suitable to on-line calculation. Experiments on an in-house developed five-axis CNC platform verify that the maximal approximation errors of both tool tip position and tool orientation are constrained, and the proposed C 3 continuous corner smoothing algorithm has higher tracking accuracy and lower acceleration frequency content at higher frequencies than the C 2 continuous algorithm.

An analytical local corner smoothing algorithm for five-axis CNC machining

Binder Jetting Process is an Additive Manufacturing technique (AM) in which a liquid binder is employed for establishing the initial strength and fabricating the geometry of components. In this process, the delivery of the binding agent is accomplished through a drop-on-demand (DOD) printhead by deposition of picoliter-sized droplets of the liquid binder. The velocity of the droplets impinging the powder bed surface might have significant effect on droplet spreading and absorption dynamics, which can be manifested in quality and integrity of the fabricated parts. In the present study, the effect of the printing speed on dimensional accuracy and equilibrium saturation level of printed samples is experimentally investigated and the observed trends are discussed in detail.

Effect of printing speed on quality of printed parts in Binder Jetting Process

This paper presents a novel kinematic corner smoothing technique for high-speed CNC machine tools. Typically, reference tool-paths compromised of short G01 moves are geometrically smoothed by means of arcs and splines within the NC system. In this study, a continuous feed motion is generated by directly planning jerk limited velocity transitions for the drives in the vicinity of sharp corners of the tool-path. This approach completely eliminates the need for geometry based path smoothing and feed planning. Contouring errors at sharp corners are controlled analytically by accurately calculating cornering speed and duration. Since proposed approach bases on kinematically planning axis motion profiles, it exploits acceleration and jerk limits of the drives and delivers a near-time optimal motion. Experimental validation and comparisons are presented to show significant improvement in the cycle time and accuracy of contouring Cartesian tool-paths.

Kinematic corner smoothing for high speed machine tools

Speed and accuracy are conflicting demands that must be met by manufacturing machines. This conflict is very frequently encountered when executing sharp corners where, to achieve tight tolerances, machine axes have to slow down considerably thus sacrificing cycle time. The most common way of reducing this tradeoff between speed and accuracy is to smooth the corner using a pre-specified curve (e.g., a spline) that allows high-speed cornering subject to path tolerance and machine kinematic limits. This paper presents a different corner-smoothing approach where the smoothing curve is not specified a priori. Instead, an optimal control method is used to generate the best free-form curve that minimizes corner traversal time while adhering to path tolerance and kinematic constraints. The resulting optimal free-form curve is parameterized using an enhanced NURBS curve fitting process to allow its accurate execution in a CNC. Illustrative examples are presented to demonstrate the merits of the proposed approach.

Minimum-time cornering for CNC machines using an optimal control method with NURBS parameterization

A limited-preview filtered B-spline approach to tracking control – With application to vibration-induced error compensation of a 3D printer

A trajectory optimization method for improved tracking of motion commands using CNC machines that experience unwanted vibration

This paper presents a method for designing a controller that achieves the best positioning performance while maximizing the energy efficiency of a redundantly actuated hybrid feed drive. A two-degree-of-freedom controller, consisting of a feedforward (FF) controller for tracking and a feedback (FB) controller for regulation, is assumed. It is shown that the ideal FF controller, which achieves perfect tracking with maximum efficiency, is not always stable. Therefore, a method for designing a stable FF controller that achieves perfect tracking with near optimal efficiency is proposed. Furthermore, the optimal relationship between FB control inputs, which guarantees maximum efficiency for any specified regulation performance, is derived. Two approaches for using the derived optimal relationship to synthesize a FB controller that achieves the best positioning performance while maximizing efficiency are proposed. Simulations and machining experiments are conducted to demonstrate the effectiveness of the proposed energy-efficient FF and FB controller design methods. Significant improvements in energy efficiency (without sacrificing positioning performance) are reported.

Energy-Efficient Controller Design for a Redundantly Actuated Hybrid Feed Drive With Application to Machining

An approach for minimizing tracking errors in linear time-invariant (LTI) single-input single-output (SISO) discrete-time systems with nonminimum phase (NMP) zeros using filtered basis functions (FBF) is studied. In the FBF method, the control input to the system is expressed as a linear combination of basis functions. The basis functions are forward filtered using the dynamics of the NMP system, and their coefficients are selected to minimize the error in tracking a given desired trajectory. Unlike comparable methods in the literature, the FBF method is shown to be effective in tracking any desired trajectory, irrespective of the location of NMP zeros in the z-plane. The stability of the method and boundedness of the control input and system output are discussed. The control designer is free to choose any suitable set of basis functions that satisfy the criteria discussed in this paper. However, two rudimentary basis functions, one in time domain and the other in frequency domain, are specifically highlighted. The effectiveness of the FBF method is illustrated and analyzed in comparison with the truncated series (TS) approximation method. [DOI: 10.1115/1.4034367]

Molong Duan

Papers

Minimum-time cornering for CNC machines using an optimal control method with NURBS parameterization

A limited-preview filtered B-spline approach to tracking control – With application to vibration-induced error compensation of a 3D printer

A trajectory optimization method for improved tracking of motion commands using CNC machines that experience unwanted vibration

Energy-Efficient Controller Design for a Redundantly Actuated Hybrid Feed Drive With Application to Machining

Tracking Control of Linear Time-Invariant Nonminimum Phase Systems Using Filtered Basis Functions