Robotics and Kinematics

Chapter 14
Industrial Integration, Performance, and Engineering Decisions

Bachelor Program in Mechatronics Engineering
Robotics and Kinematics   •   Bachelor

Industrial Integration, Performance, and Engineering Decisions

Introduction

Industrial robotics is no longer an isolated activity in which a robot simply repeats a programmed motion. A modern robotic cell is a cyber-physical production system: mechanical equipment, sensors, actuators, controllers, safety systems, databases, operators, and manufacturing-planning software must function as one coordinated system. The engineering challenge is therefore not merely to select a robot with sufficient payload and reach. It is to design a safe, productive, maintainable, economically justified, and information-connected workflow.

Integration decisions influence product quality, production capacity, labor requirements, energy consumption, cybersecurity exposure, and the ability of a factory to respond to changing demand. This chapter develops a systems approach to robotic-cell design and evaluates performance through accuracy, repeatability, cycle time, throughput, reliability, maintainability, energy, cost, return on investment, and data-driven improvement.

Learning objectives

By the end of this chapter, a student should be able to:

  1. Translate a manufacturing requirement into a robotic-cell concept and workflow.
  2. Explain the roles of PLCs, robot controllers, machine vision, industrial networks, IIoT, and manufacturing-execution systems.
  3. Distinguish robot accuracy from repeatability and select appropriate performance measures.
  4. Calculate cycle time, capacity, availability, performance, quality, throughput, and basic economic indicators.
  5. Analyze reliability, maintainability, energy use, and lifecycle cost.
  6. Make engineering decisions under technical, economic, safety, ethical, and organizational constraints.

1. From task description to robotic-cell architecture

A successful cell begins with a precise statement of the manufacturing problem. The specification should identify the product family, operations, takt requirement, part variation, quality criteria, material flow, operator interaction, environmental conditions, available utilities, and applicable safety requirements.

A useful design sequence is:

  1. Define the process: What transformation, inspection, handling, assembly, or packaging operation is required?
  2. Characterize the workpiece: Determine mass, dimensions, material, surface condition, allowable contact force, tolerances, and presentation method.
  3. Quantify demand: Establish required production rate, shift pattern, batch size, changeover frequency, and expected product mix.
  4. Select the process strategy: Compare manual, dedicated automation, robot-based automation, and hybrid alternatives.
  5. Allocate functions: Decide which functions belong to the robot, PLC, vision system, safety controller, drives, supervisory computer, or operator.
  6. Design material and information flow: Specify infeed, buffering, workholding, inspection, rework, outfeed, and data exchange.
  7. Validate: Check reachability, collision avoidance, cycle time, safety, ergonomics, maintainability, and economic feasibility before procurement.

A cell normally includes a robot manipulator and end-of-arm tooling, fixtures, conveyors or feeders, sensors, a PLC, safety devices, a human–machine interface, and communication interfaces. The robot controller executes motion instructions, while the PLC commonly coordinates sequencing, interlocks, peripheral equipment, and machine states. This separation is not universal: some integrated platforms combine functions, but the responsibilities must remain clearly defined.

Workflow and state-based control

A robust workflow is easier to design when represented as states rather than as an unstructured list of commands. Typical states include idle, initialization, part request, part present, grip confirmation, motion, process execution, inspection, unload, fault, and recovery. Transitions should be triggered by verified conditions, not by assumed timing.

For example, a robot should not leave an infeed position merely because a timer has elapsed. It should receive a valid part-present signal, confirm that the gripper is closed and pressure is adequate, and verify that the safety system permits motion. State-based design improves troubleshooting because each failure can be associated with a state and a missing transition condition.

A simplified cell sequence is:

Integrated robotic cell workflow Part arrivessensor / vision PLC verifiesinterlocks Robot picksand transfers Process / inspectquality decision Unload and logtraceability Fault, reject, or recovery path
Integrated robotic cell workflow
Figure: Integrated robotic cell workflow

Figure: A state-oriented workflow for an integrated robotic cell.

2. Integration of control, sensing, and information systems

PLC and robot controller

The PLC is generally responsible for deterministic machine coordination: sequencing peripheral devices, reading sensors, controlling valves and conveyors, managing alarms, and enforcing permissive conditions. The robot controller manages interpolation, joint and Cartesian motion, tool data, frames, speed limits, and robot-specific diagnostics. Reliable integration requires an explicit interface list, signal definitions, timing expectations, fault behavior, and reset philosophy.

Important handshake signals may include cell ready, robot ready, cycle start, part available, gripper closed, process complete, inspection pass, inspection fail, and fault acknowledged. Each signal needs a defined source, destination, meaning, normal state, abnormal state, and recovery action. Ambiguous handshakes are a major cause of intermittent faults.

Machine vision

Vision systems may locate parts, identify variants, measure dimensions, verify assembly, read codes, or detect defects. A typical architecture includes illumination, optics, camera, image-processing software, calibration data, and a communication link to the PLC or robot.

The engineering decision is not simply whether vision can detect an object. The designer must ask whether lighting remains stable, whether the required measurement uncertainty is acceptable, how false positives and false negatives are handled, and what happens when confidence is low. Vision results should normally include a validity or confidence status rather than a bare pass/fail value. Calibration must relate pixel coordinates to machine or robot coordinates, and recalibration procedures must be defined after camera, fixture, or tooling changes.

Industrial networks

Industrial Ethernet and fieldbus systems connect controllers, drives, remote I/O, safety devices, vision equipment, and supervisory computers. Selection depends on real-time requirements, topology, determinism, interoperability, diagnostic capability, safety communication, installed skills, and lifecycle support. A network architecture should separate control traffic from enterprise traffic where appropriate and should document device identities, addressing, update procedures, and failure modes.

A communication link is not automatically safe merely because it is digital. Functional safety requires an appropriate safety architecture, validated safety functions, and compliance with applicable standards and risk-assessment procedures. Emergency stops, guard interlocks, safe torque off, speed limitation, and enabling devices must be designed as safety functions rather than ordinary software conditions.

IIoT and manufacturing systems

Industrial Internet of Things (IIoT) systems collect machine data for monitoring, traceability, predictive maintenance, and optimization. A practical information hierarchy is:

  • Field level: sensors, actuators, encoders, and safety devices.
  • Control level: PLCs, robot controllers, motion controllers, and HMIs.
  • Supervisory level: historians, dashboards, alarm servers, and supervisory control.
  • Manufacturing level: manufacturing-execution systems (MES), quality databases, scheduling, and genealogy.
  • Enterprise level: enterprise-resource planning (ERP), supply-chain, financial, and customer systems.

Data should have a purpose. Useful tags may include cycle start and end times, recipe, product identifier, robot program version, fault code, inspection result, energy value, and maintenance event. Poorly defined tags, unsynchronized clocks, missing context, and uncontrolled software changes can make a large data set nearly useless.

A basic traceability record might link a product serial number to the material lot, station, operator or authorized user, robot program, tooling identifier, inspection measurements, and disposition. Such records support quality investigation but also introduce responsibilities concerning access control, privacy, retention, and cybersecurity.

3. Measuring robotic-cell performance

Accuracy and repeatability

Accuracy is the closeness of a robot’s achieved position to its commanded or nominal position. Repeatability is the closeness of repeated achieved positions to one another. A robot can be highly repeatable but inaccurate if its calibration or work-frame definition is wrong. In production, repeatability may be more important for a fixed fixture, whereas absolute accuracy is critical for operations involving independently located parts or coordinated machines.

For measured positions, the mean position is:

x̄ = (1/n) ∑ᵢ₌₁ⁿ xᵢ

where xi is the measured coordinate vector for trial i, n is the number of trials, and x̄ is the mean coordinate vector. If the nominal target is xt, the mean positional error is:

e = x̄ − xt

The vector magnitude ||e|| expresses the Euclidean positional bias, provided all coordinates use the same reference frame and units. Repeatability can be characterized by the spread of the measurements, commonly using standard deviation. For one coordinate:

s = √[ ∑ᵢ₌₁ⁿ (xᵢ − x̄)² ÷ (n − 1) ]

This statistical estimate assumes independent, comparable measurements and a stable process. It does not by itself prove compliance with a specification; measurement-system uncertainty, temperature, payload, speed, orientation, and direction of approach must also be considered.

Cycle time, takt time, and throughput

Cycle time is the elapsed time required to complete one unit or one repeating work cycle. Takt time represents the available production time per period divided by customer demand for that period:

ttakt = Tavailable ÷ D

where Tavailable is net available production time and D is required demand, both expressed over the same period. A cell generally needs an effective cycle time no greater than takt time, after considering yield, downtime, and planned losses.

For a serial system with effective cycle time tc, ideal capacity over available time is:

Qideal = Tavailable ÷ tc

This is an ideal upper bound. If availability is A, performance efficiency is P, and quality yield is Q, the overall equipment effectiveness (OEE) convention is:

OEE = A × P × Q

Here, A, P, and Q are fractions between 0 and 1. Availability is commonly planned production time minus downtime, divided by planned production time. Performance compares the ideal cycle rate with the actual operating rate, and quality is good units divided by total units. Organizations must define time categories consistently because different definitions produce incomparable OEE values.

Worked example. A cell is planned for 450 minutes per shift. Downtime is 36 minutes, the ideal cycle time is 18 seconds per piece, 1,320 pieces are produced, and 30 are rejected. Planned production time is 27,000 seconds and operating time is 24,840 seconds. Thus:

  • Availability = 24,840 ÷ 27,000 = 0.920.
  • Performance = (18 × 1,320) ÷ 24,840 = 0.9565.
  • Quality = (1,320 − 30) ÷ 1,320 = 0.9773.
  • OEE = 0.920 × 0.9565 × 0.9773 ≈ 0.860, or 86.0%.

The result is not a diagnosis. The three factors must be examined separately: a high OEE can coexist with an unsafe design, excessive energy use, or a bottleneck elsewhere in the factory.

Bottlenecks and line balance

For serial stations, the bottleneck is usually the station with the greatest effective cycle time. If parallel identical stations operate independently and share demand evenly, ideal capacity is approximately the sum of their capacities, but buffers, synchronization, changeovers, and quality constraints reduce this value. A line should be evaluated as a system rather than by optimizing one robot in isolation.

4. Reliability, maintainability, and energy

Reliability is the probability that an item performs its required function for a specified time under stated conditions. For a constant failure rate λ, an exponential reliability model is:

R(t) = e^(−λt)

where t is operating time and λ has units of inverse time. The model assumes a constant hazard rate and is often a rough approximation during the useful-life period; it is not universally valid during early-life failures or wear-out.

Mean time between failures (MTBF) is commonly estimated as operating time divided by the number of relevant failures. Mean time to repair (MTTR) is repair time divided by the number of repairs. Under simplified steady-state assumptions, inherent availability can be approximated by:

Ai = MTBF ÷ (MTBF + MTTR)

This expression excludes logistical delays, planned maintenance, starvation, blocking, and quality losses. Operational availability should include the delays relevant to the actual production system.

Maintainability is improved by accessible components, diagnostic messages, standardized connectors, modular tooling, safe isolation procedures, spare-parts planning, and clear recovery instructions. A sophisticated predictive-maintenance model is of little value if technicians cannot safely access the failed component or if replacement parts have long lead times.

Energy analysis should distinguish operating energy, standby energy, compressed-air consumption, process energy, and embodied energy where lifecycle analysis is intended. A simple specific-energy indicator is:

Especific = Etotal ÷ Ngood

where Etotal is energy consumed over a defined period and Ngood is the number of conforming units produced in that period. The boundary and measurement interval must be stated. Reducing cycle time may increase peak power; reducing robot speed may lower energy per cycle but reduce throughput. Decisions should therefore compare energy per good part, not only instantaneous power.

5. Cost, investment, and engineering economics

A robotic project includes more than robot purchase price. Capital cost may include robot, tooling, fixtures, conveyors, vision, safety equipment, controls, installation, integration, validation, training, and infrastructure. Operating cost includes energy, consumables, maintenance, software support, scrap, downtime, and labor. Lifecycle cost is often more informative than initial price.

A simple payback period is:

Payback period = Initial investment ÷ Annual net cash benefit

The formula assumes a reasonably stable annual benefit and ignores the time value of money. Annual net cash benefit should include recurring savings minus recurring additional costs, not gross production value alone.

For projects with unequal cash flows, net present value is more appropriate:

NPV = −C₀ + ∑ₜ₌₁ⁿ [CFₜ ÷ (1 + r)ᵗ]

where C0 is initial investment, CFt is net cash flow in period t, r is the discount rate per period, and n is the number of periods. A positive NPV indicates that the forecast cash flows exceed the required return under the chosen assumptions. Sensitivity analysis should vary demand, utilization, labor cost, scrap reduction, maintenance cost, and project delay.

Economic decisions should also account for flexibility. A dedicated machine may have lower unit cost at stable high volume, while a robot may offer greater product variety and easier reconfiguration. Flexibility has value when product mix or demand is uncertain, but it should not be claimed without identifying the avoided future cost or risk.

6. Data-driven optimization and artificial intelligence

Optimization begins with trustworthy measurement. A disciplined improvement loop is:

  1. Define a performance objective, such as good parts per hour or energy per good part.
  2. Establish a baseline using consistent time and quality definitions.
  3. Collect contextualized data, not isolated raw signals.
  4. Identify losses using Pareto analysis, correlation, process mapping, or designed experiments.
  5. Test a controlled change.
  6. Validate that improvement is statistically and operationally meaningful.
  7. Standardize the change and monitor for drift.

Artificial intelligence can support vision inspection, anomaly detection, remaining-useful-life estimation, scheduling, and parameter recommendation. However, an AI model can reproduce biased labels, fail under changed lighting or product variants, and generate false confidence. Models should be validated on representative data, monitored after deployment, and provided with a safe fallback. A model should recommend or classify only within a defined operating domain unless its safety implications have been rigorously addressed.

A useful distinction is between descriptive analytics—what happened; diagnostic analytics—why it happened; predictive analytics—what is likely to happen; and prescriptive analytics—what action should be taken. A dashboard is not automatically an optimization system. Human expertise remains essential for causal interpretation and safe intervention.

7. Safety, cybersecurity, ethics, and governance

Risk assessment should consider hazards from robot motion, crushing and shearing, unexpected restart, sharp tooling, stored pneumatic or electrical energy, hot processes, ejected parts, ergonomic exposure, and software-induced behavior. Protective measures follow a hierarchy: inherently safer design, safeguarding, engineering controls, administrative procedures, training, and personal protective equipment. Safety validation must address foreseeable misuse and recovery operations, not only normal automatic production.

Cybersecurity is part of functional availability and safety. Important practices include network segmentation, least-privilege accounts, secure remote access, asset inventories, tested backups, patch governance, removable-media control, event logging, and recovery exercises. Security controls must be coordinated with real-time and safety requirements; indiscriminate updates can themselves interrupt production.

Ethical engineering includes honest reporting of performance, transparent treatment of worker displacement, protection of worker data, and avoidance of surveillance beyond legitimate operational purposes. Automation should be evaluated for its effect on job quality, training, accessibility, and human dignity. Regulatory obligations depend on jurisdiction and application, but designers should consult current machinery-safety, electrical, robot, occupational-health, data-protection, and cybersecurity requirements rather than relying on outdated checklists.

8. Illustrative case: vision-guided packaging cell

Background

A food-equipment manufacturer packages three variants of a small actuator. Operators currently identify variants visually, place them into cartons, and record quantities manually. Demand is 1,200 good units per eight-hour shift. Breaks and planned meetings leave 420 minutes of planned production time.

Problem

Manual identification causes mislabeling and variable output. Management proposes a six-axis robot, a camera, automatic carton indexing, and MES traceability. The project team must determine whether the concept meets demand and whether the investment is justified.

Analysis

The takt time is:

ttakt = 420 × 60 ÷ 1,200 = 21 s/unit

The proposed sequence requires 3 s for vision acquisition and decision, 8 s for pick and placement, 5 s for carton indexing and confirmation, and 2 s for data logging, giving a nominal 18 s cycle. Because the process is not perfectly available, a nominal margin of only 3 seconds is potentially inadequate. The team therefore studies historical stoppages and estimates availability of 0.93 and quality yield of 0.99. If the 18-second cycle is maintained, expected good output is approximately:

Qgood ≈ (420 × 60 ÷ 18) × 0.93 × 0.99 ≈ 1,279 units/shift

This exceeds demand, but the margin is modest. A buffer for cartons and a reject lane are added so that a vision failure does not stop the entire line. The PLC controls carton indexing and interlocks; the robot controller executes motion; the vision system returns variant, pose, confidence, and validity; and the MES records serial or batch data.

The team also tests reflective surfaces, variation in part orientation, camera contamination, and recovery after a rejected image. A low-confidence result is diverted for manual verification rather than being automatically accepted.

Economic decision

The installed cost is estimated at $180,000. Annual net benefit is forecast at $72,000 after maintenance, energy, software support, and training. Simple payback is:

180,000 ÷ 72,000 = 2.5 years

A sensitivity analysis shows that a 20% reduction in demand would extend payback substantially, while a 3% improvement in quality and reduced relabeling would improve it. The project is approved conditionally, subject to a pilot demonstrating the required detection rate and a validated safety assessment.

Outcome and learning points

The pilot demonstrates 1,310 good units per shift, but only after optimizing lighting and adding a carton buffer. The final design includes scheduled lens cleaning, a spare camera, controlled recipe management, and operator training for recovery. The principal lesson is that robot selection alone did not determine success. Workflow design, quality logic, data context, maintainability, and economic assumptions were equally important.

9. Best-practice decision framework

A professional design review should ask:

  • Does the proposed cell meet takt and quality requirements under realistic availability?
  • Is the bottleneck understood, and can it be relieved without transferring the problem downstream?
  • Are robot, PLC, vision, safety, and MES responsibilities unambiguous?
  • Are accuracy, repeatability, calibration, and measurement uncertainty appropriate to the process?
  • Can an operator safely start, stop, inspect, clear, and recover the cell?
  • Can a maintainer diagnose faults and replace critical components efficiently?
  • Are energy and lifecycle costs measured using clear boundaries?
  • Are cybersecurity, software versioning, backups, and change control included?
  • Is the investment robust to demand, labor, quality, and downtime uncertainty?
  • Does the design improve the whole manufacturing system rather than one local metric?

Challenges and limitations

Robotic integration is limited by uncertain product variation, imperfect sensors, calibration drift, network failures, supplier-specific software, incomplete production data, and changing regulations. Performance metrics can be gamed when definitions are unclear—for example, by excluding minor stops or counting reworked units as good output. Simulation can reveal reach and cycle-time problems, but it cannot fully predict operator behavior, contamination, component wear, or organizational resistance. Economic forecasts are also sensitive to utilization and hidden integration costs. Consequently, commissioning should include staged tests, realistic pilot runs, documented acceptance criteria, and post-deployment review.

Chapter Summary

Industrial robotic engineering is a systems activity involving mechanics, controls, sensing, safety, information technology, people, and economics. A cell should be designed from the manufacturing requirement outward, with explicit workflows, state-based sequencing, defined PLC–robot handshakes, robust vision decisions, suitable industrial networks, and purposeful IIoT data.

Accuracy and repeatability are different properties; cycle time must be compared with takt time; and throughput must account for availability, performance, and quality. Reliability and maintainability determine whether nominal capacity can be sustained. Energy should be evaluated per good unit, while lifecycle economics should include integration, operation, maintenance, and flexibility. NPV and sensitivity analysis provide stronger investment decisions than simple purchase-price comparison. Finally, safety, cybersecurity, ethical treatment of workers, data governance, and regulatory compliance are integral engineering requirements, not optional additions.

Key Terms

Accuracy: Closeness of an achieved position to its nominal target.

Repeatability: Closeness of repeated measurements or robot positions to one another.

PLC: Programmable logic controller used for deterministic machine coordination.

Robot controller: Controller responsible for robot motion, frames, tools, and programs.

Machine vision: Use of cameras and image processing for identification, measurement, or inspection.

Industrial network: Communication system connecting automation devices and controllers.

IIoT: Industrial Internet of Things; connected sensing and data systems for industrial purposes.

MES: Manufacturing-execution system coordinating production information and traceability.

Takt time: Available production time divided by required demand.

Cycle time: Time required to complete one repeating production cycle.

Throughput: Rate of completed production, usually expressed as units per time.

Bottleneck: System element that limits overall flow or capacity.

OEE: Product of availability, performance, and quality factors.

Reliability: Probability of performing a required function for a specified time.

Maintainability: Ease and speed with which an item can be restored after failure.

MTBF: Mean time between relevant failures.

MTTR: Mean time to repair.

Specific energy: Energy consumed per good unit produced.

Net present value: Present value of future net cash flows minus initial investment.

Traceability: Ability to relate a product to its process, material, inspection, and configuration history.

Review Questions

Short Answer Questions

  1. Distinguish the principal responsibilities of a PLC and a robot controller.
  2. Why can a robot be repeatable but inaccurate?
  3. Define takt time and explain how it differs from cycle time.
  4. What information should be included in a reliable PLC–robot handshake?
  5. Why is energy per good unit often more useful than instantaneous power?

Long Answer Questions

  1. Explain a systematic method for designing an integrated robotic cell from process requirement through validation.
  2. Discuss how machine vision, industrial networks, IIoT, and MES can be combined to support quality and traceability.
  3. Explain OEE, including its three components, assumptions, and limitations.
  4. Compare simple payback and NPV as methods for evaluating a robotic investment.
  5. Discuss how safety, cybersecurity, maintainability, and ethics should influence cell architecture.

Analytical / Application Questions

  1. A cell has 480 planned minutes, 40 minutes of downtime, an ideal cycle time of 20 seconds, 1,250 total units, and 25 rejects. Calculate availability, performance, quality, and OEE using the definitions in this chapter.
  2. A proposed cell has a 16-second nominal cycle, but estimated availability is 0.90 and quality is 0.98. Planned production time is 400 minutes and demand is 1,350 good units. Determine whether the estimated good output meets demand and identify at least two design responses if it does not.
  3. A robotic project costs $240,000 and is expected to generate annual net benefits of $80,000, but demand may vary by ±25%. Analyze the simple payback and explain why a sensitivity analysis and NPV calculation would be needed before approval.

Multiple Choice Questions with Answers

Question 1

Which function is most commonly assigned to a PLC in an integrated robotic cell?

A. Interpolating the robot’s joint trajectory

B. Coordinating peripheral equipment and interlocks

C. Calibrating camera pixels automatically in every case

D. Calculating the company’s annual depreciation

Correct Answer: B. Coordinating peripheral equipment and interlocks

Question 2

A robot repeatedly reaches nearly the same location, but that location is offset from the commanded target. This indicates high repeatability but poor what?

A. Availability

B. Accuracy

C. Maintainability

D. Throughput

Correct Answer: B. Accuracy

Question 3

If 420 minutes are available to produce 1,200 units, what is the takt time?

A. 14 s/unit

B. 18 s/unit

C. 21 s/unit

D. 28 s/unit

Correct Answer: C. 21 s/unit

Question 4

Using OEE = availability × performance × quality, what is the conceptual purpose of OEE?

A. To measure only robot positioning error

B. To combine major production losses into one indicator

C. To determine the discount rate for investment

D. To replace a functional safety assessment

Correct Answer: B. To combine major production losses into one indicator

Question 5

Which condition is required for the simple exponential reliability model R(t) = e^(−λt) to be a reasonable approximation?

A. A constant failure rate over the modeled interval

B. Zero operating time

C. A guaranteed absence of maintenance

D. A changing failure rate caused by wear-out

Correct Answer: A. A constant failure rate over the modeled interval

Question 6

A cell consumes 600 kWh and produces 1,500 good units during a measurement period. What is its specific energy?

A. 0.4 kWh per good unit

B. 2.5 kWh per good unit

C. 900 kWh per good unit

D. 2,100 kWh per good unit

Correct Answer: A. 0.4 kWh per good unit

Question 7

Which practice most directly improves traceability?

A. Recording only the robot brand

B. Linking product identity to recipe, inspection, and process records

C. Removing timestamps to reduce database size

D. Allowing uncontrolled program changes

Correct Answer: B. Linking product identity to recipe, inspection, and process records

Question 8

Why should a low-confidence vision result normally be handled explicitly?

A. A camera always provides a legally certain answer

B. It prevents uncertain classification from being silently accepted

C. It eliminates the need for lighting design

D. It guarantees zero false negatives

Correct Answer: B. It prevents uncertain classification from being silently accepted

Question 9

An initial investment is $180,000 and annual net cash benefit is $72,000. What is the simple payback period?

A. 1.5 years

B. 2.0 years

C. 2.5 years

D. 3.5 years

Correct Answer: C. 2.5 years

Question 10

Which statement best describes NPV?

A. It ignores the timing of cash flows

B. It discounts future net cash flows and subtracts initial investment

C. It measures only robot repeatability

D. It is identical to annual energy consumption

Correct Answer: B. It discounts future net cash flows and subtracts initial investment

Robotics and Kinematics
Chapter 14 of 15